Download An introduction to lattice gauge theory and spin systems

to lattice gauge theory and spin systerais"
An introduction
John B. Kogut
Department
of Physics,
Uniuersity
of Illinois at Urbana-Champaign,
Urbana, Illinois 61801
This article is an interdisciplinary
review of lattice gauge theory and spin systems. It discusses the
both physics and formalism, of these related subjects. Spin systems are models of
fundamentals,
magnetism and phase transitions. Lattice gauge theories are cutoff formulations of gauge theories of
strongly interacting particles. Statistical mechanics and field theory are closely related subjects, and the
connections between them are developed here by using the transfer matrix. Phase diagrams and critical
the character and continuum
points of continuous transitions are stressed as the keys to understanding
limits of lattice theories. Concepts such as duality, kink condensation, and the existence of a local,
relativistic field theory at a critical point of a lattice theory are illustrated in a thorough discussion of the
two-dimensional
Ising model. Theories with exact local (gauge) symmetries are introduced following
%'egner's Ising lattice gauge theory. Its gauge-invariant "loop" correlation function is discussed in detail.
Three —dimensional Ising gauge theory is studied thoroughly. The renormalization group of the two
dimensional planar model is presented as an illustration of a phase transition driven by the condensation
of topological excitations. Parallels are drawn to Abelian lattice gauge theory in four dimensions. NonAbelian gauge theories are introduced and the possibility of quark confinement is discussed. Asymptotic
freedom of O(n) Heisenberg spin systems in two dimensions is verified for n
and is explained in .
simple terms. The direction of present-day research is briefly reviewed.
)
C. The planar
CONTENTS
—
Introduction
An Overview of this Article
Phenomenology and Physics of Phase Transitions
A. Facts about critical behavior
B. Correlation length scaling and the droplet
picture
659
661
661
—
The Transfer Matrix Field Theory and Statistical Mechanics
A. General remarks
B. The path integral and transfer matrix of the
s imple harmonic oscillator
C. The transfer matrix for field theories
The Two-Dimensional Ising Model
A. Transfer matrix and 7-continuum formulation
B. Self-duality of the Ising model
C. Strong coupling expansions for the mass gap,
weak coupling expansions for the magnetization
D. Kink condensation and disorder
E. Self-duality of the isotropic Ising model
F. Exact solution of the Ising model in two dimensions
V. Wegner's Ising Lattice Gauge Theory
A. Global symmetries, local symmetries, and
the energetics of spontaneous symmetry
breaking
B. Constructing an Ising model with a local
symmetry
C. Elitzur 's theorem the impossibility of spontaneously breaking a local symmetry
D. Gauge-invar iant correlation functions
E. Quantum Hamiltonian and phases of the threedimensional Ising gauge theory
VI. Abelian Lattice Gauge Theory
A. General formulation
B. Gauge-invariant correlation functions, physical interpretations, and phase diagrams
C. The quantum Hamiltonian formulation and
quark confinement
The Planar Heisenberg Model in Two Dimensions
A. Introductory comm. ents and motivation
B. The physical picture of Kosterlitz and
Thouless
—
Supported in part by the National Science Foundation
grant number NSF PHY77-25279.
Reviews of Modern Physics, Vol.
51, No. 4, October 1979
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model in the periodic Gaussian
approximation
D. Renormalization group analysis and the theory's critical region
VIII. Non-Abelian Lattice Theories
A. General formulation of the SU(2) theory
B. Special features of the non-Abelian theory
C. Renormalization group analysis of O(n) spin
systems in two dimensions
D. Results from the Migdal recursion relation
IX. Parting Comments
Acknowledgment
References
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under
I. INTRODUCTION
—AN
OVERVIEW OF THIS ARTICLE
This article consists of a series of introductory lectures on lattice gauge theory and spin systems. It is
intended to explain some of the essentials of these subjects to students interested in the field and research
physicists whose expertise lies in other domains. The
expert in lattice gauge theory will find little new in the
following pages aside from the author's personal perspective and overview. The style of this presentation
is informal. The article grew out of a half-semester
-graduate course on lattice physics presented at the University of Illinois during the fall semester of 1978.
Lattice spin systems are familiar to most physicists
because they model solids that are studied in the laboratory. These systems are of considerable interest in
these lectures, but we shall also be interested in more
abstract questions. In particular, we shall be using
space-time lattices as a technical device to define cutoff field theories. The eventual goal of these studies is
to construct solutions of cutoff theories so that field
theories defined in real continuum Minkowski spacetime can be understood. The lattice is mere scaffolding an intermediate step used to analyze a difficult
nonlinear system of an infinite number of degrees of
freedom. Different lattice formulations of the same
—
Copyright
) 979 American Physical Society
660
John B. Kogut: Lattice gauge theory and spin systems
field theory can be contemplated, just
crete versions of differential equations
down. Space —time symmetric la, ttices
dimensions will be discussed in detail
quite el. egant. Another approach which
as different dis-
can be written
in four Euclidean
because they are
leaves the
"time" axis continuous and replaces continuum spatial
axes by a three-dimensional lattice will also be dis-
cussed.
Once a lattice field theory has been formulated, the
original field theory problem becomes one of statistical
mechanics. This point will be developed in detail in this
article through both general analyses and specific examples. The first step in understanding the theory is
then to map out the phase diagram of the equivalent statistical mechanics system. The ground state of the theory changes qualitatively from one phase to another. In
this review we shall be particularly interested in lattice
gauge theories which model the strong interactions of
particle physics. In some of these theories there is a
range of parameters for which the ground state cannot
tolerate the presence of an isolated quark. This is the
quark-confining phase of the cutoff theory. This phase
may be separated by a critical surface from another
phase in which quarks can be isolated. After the phases
of the theory have been established, the behavior of the
One
theory in the critical region must be determined.
must determine, for example, the order of the transition occurring between the two phases. Only if the transition is continuous can one obtain a continuum, relativistic field theory from the lattice system. If the
transition is continuous, the system's mass gap vanishes as a critical point is approached. One can then
define a renormalized mass of the field theory, which
can be held fixed as the lattice spacing is taken to zero.
At the critical. point the theory loses memory of the lattice and the continuous space —time symmetries of the
field theory are reestablished.
These points will be
discussed at length throughout this article. In short, to
use lattice formulations of field theory to construct continuum theories, one maps out the lattice theory's phase
diagram, locates its critical points (lines or surfaces)
of continuous phase transitions, and approaches the
critical. points in a mell-defined, delicate fashion.
The first step in this program is the least difficult.
The phases of a statistical mechanics system can usually be established using several methods of varying reliability: high- or low-temperature expansions, duality
transformations,
spin-wave analyses, mean field theory, etc. In particular cases one must use these techniques judiciously. If a system has complicated symmetries, it can be difficult indeed to establish its phase
diagram with confidence. Frequently thorough numerical calculations using expansion methods. or renormalization groups are required. These same methods can
then be used to determine the nature of the system's
critical points. In particular, at a continuous phase
transition various thermodynamic functions of the system become singular and the degree of their singularities is recorded in their critical exponents. We shall
illustrate the calculation of some of these exponents for
model lattice spin systems in later chapters. The twodimensional Ising model will also be solved in closed
form to illustrate the fact that at its critical point it
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
defines a relativistic, scale-invariant field theory.
Lattice gauge theories pose special problems to this
traditional approach to studying spin systems. Ising
gauge theories were invented in a remarkable article
by F. Wegner in 1971 (Wegner, 1971). He elevated the
down symmetry of the ordinary Ising model
global up
to a local symmetry of his new theory's action. He
realized that such a model could undergo phase transitions, but they could not be accompanied by a spontanThe absence of a local magnetizaeous magnetization.
tion challenged him to find a sensible way to label the
phases of the theory. Wegner stressed the importance
of correlation functions here. In fact, he was forced to
construct a correlation function which respected the local up
down symmetry of the model. This led him to
the "loop correlation function" and the "area" and "perimeter" laws which will be discussed in Sec. V.
There are two phenomena which play an especially
important role in the physics of lattice gauge theories.
The first is the occurrence of topological excitations,
which can label the, phases of some of these models and
determine the character of their ground states. The
second is asymptotic freedom and the possibility that
non-Abelian gauge theories exist only in a single,
quark-confining phase. The importance of topological
excitations will be illustrated in the two-dimensional
Ising model, the planar model, and four-dimensional
Abelian lattice gauge theory. Topological excitations
are stationary configurations of the theory's action.
They affect the local variables of the theory over an infinite domain of space and tend to disorder the system.
These models' high-temperature phases can be described as condensates of such excitations. These facts
will be illustrated in a renormalization group calculation of the planar model's phase diagram (Kosterlitz,
1974). It is believed that topological excitations which
resemble dynamical magnetic monopoles play an important role in determining the character of the ground
state of non-Abelian gauge theories in four dimensions
('t Hooft, 1978).
The discovery that non-Abelian gauge theories in four
dimensions are asymptotically free ('t Hooft, 1972;
Politzer, 1973; Gross and Wilczek, 1973) has profoundly affected the modern study of field theories.
Roughly speaking, asymptotic freedom means that lower- frequency fluctuations in a theory are more strongly
If the thecoupled than higher- frequency fluctuations.
ory were formulated with a momentum space cutoff A
and a coupling constant g(A), then another formulation
using a small. er cutoff A' would need a larger coupling
g(A') to describe the same physics at a given physical
momentum sca. le. As A- ~, the coupling g(A) vanishes
and the theory is free at short distances. The fact that
g(A) vanishes as A grows allows one to make precise
predictions for deep-inelastic scattering of electrons or
neutrinos off strongly interacting particles such as protons and neutrons. These calculations compare extremely well with experiment and strongly support the
conjecture that the underlying theory of strong interactions is based on an SU(3) "color" gauge group with
quarks residing in the fundamental representation of the
group (Fritzsch et a/. , 1973). It is this experimental
success that has made gauge theories of strong inter-
=
=
John B. Kogut: Lattice gauge theory and spin systems
actions so attractive to the high-energy
physics com-
munity.
Immediately after the discovery of asymptotic freedom, it was conjectured by innumerable theorists t.hat
quarks would be confined in such theories. Asymptotic
freedom suggests the possibility that the coupling g(A)
grows large as A decreases, so that it would be energetically favorable for quarks to bind together into color-singlet bound states rather than to exist alone. This
idea could then reconcile the great success of nonAbelian gauge theories in describing deep-inelastic
phenomena with the absence of free quarks in the de-
bris of these scattering events. It is easy to establish
that lattice gauge theory confines quarks at strong coupling (Wilson, 1974). As in many spin systems, reliable calculations can be made in lattice gauge theories
in the strong coupling (high-tempera, ture) doma. in.
From this point of view, however, the problem of obtaining a theory of strong interactions becomes one of
establishing asymptotic freedom of the lattice theory
formulated on a fine lattice with vanishingly small coupling constant and ihe absence of any critical points at
finite coupling. Doing this would then constitute the
first step toward making a computable theory of strong
interactions in which asymptotic freedom and confined
quarks could coexist. The lattice theory of colored
quarks and gluons has not been analyzed in enough detail to decide whether this grand hope is realized. However, analogous behavior has been found in simpler
systems. Later in this review we shall see that O(n)
Heisenberg spin systems are asymptotically free (Polyakov, 1975a). It is also easy to see that these spin systems are disordered at large coupling in the sense that
their spin-spin correlation function is short ranged.
This feature of the theory is analogous to the quarkconfining property of lattice gauge theory at strong coupling. This point will be discussed at. length in Sec. VIII.
In addition, recent work strongly suggests that the O(z)
Heisenberg systems have well-behaved scattering matrices describing massive particles (Zamolodchikov
and Zamolodchikov,
1978). This indicates that such
models exist only in a disordered phase and that they
possess a well. -defined continuum limit given a relativistic, massive, interacting field theory. This model
calculation constitutes an important, optimistic step in
the lattice gauge theory program.
This article is organized as follows. First we review
some phenomenology concerning phase transitions.
Then the transfer matrix is discussed in detail and the
connection between field-theoretic and statistical mechanics language is established. Statistical mechanics
systems formulated on symmetric space-time lattices
and very anisotropic lattices in which one axis (time) is
left continuous are discussed. These general remarks
are illustrated in a detailed discussion of the Ising model. Strong coupling expansions for the theory's critical
behavior are illustrated. The self-duality of the model
is established using both the partition function for the
theory formulated on a symmetric lattice and the operator transfer matrix formulated on a lattice with one
axis continuous. Kink condensation is discussed and the
time-continuum version of the model is solved exactly
to establish that the theory. at the critical point becomes
Rev. Mod. Phys. , Vol. 6't, No.
4, October 1979
—
a scale-invariant, relativistic field theory a free,
massless fermion. Next, global and local symmetries
and spontaneous symmetry breaking are discussed in
preparation for the introduction of Ising lattice gauge
theory. F. Wegner's Ising lattice gauge theory is introduced and the fact that its mean magnetization must
vanish at all coupling is proved. The loop correlation
function is introduced and the perimeter law at small
coupling and the area law at strong coupling are estabmethods. The two-dil. ished using reliable expansion
mensional gauge model is shown to be equivalent to the
ordinary one-dimensional Ising model which exists only
formulation
in a disordered phase. The time-continuum
of the theory is developed. The three-dimensional
gauge theory is shown to be dual to the three-dimensiona) Ising model. This exercise sheds light on the
phases of the gauge theory and kink condensation.
Abelian lattice gauge theory is introduced next. The
physical interpretation of the. loop correlation function
in terms of quark confinement is derived. The energetics and phases of the model are discussed by evaluating the loop correlation function at weak coupling
where Coulomb's law emerges and at strong coupling
where a linear potential confines quarks. The timecontinuum formulation of the theory and the notion of
flux tubes are discussed. We then turn to a detailed
renormalization group analysis of the two-dimensional
planar model to see an example of a phase transition
without a local order parameter.
The Kosterlitz-Thouless physical picture of vortices and the planar model
phase transition is developed in detail (Kosterlitz and
Thouless, 1973), and the renormalization group trajectories are found'from the sine-Gordon representation
of the theory's action. The vortex structure of the twodimensional planar model is compared with the vortexloop structures of the four-dimensional Abelian lattice
gauge theory (Banks et a/. , 1977) Non-. Abelian lattice
gauge theories are discussed next. The SU(2) theory is
formulated and its special properties are discussed.
Analogies to the two-dimensional O(4) Heisenberg model
are drawn, and it is shown that O(n) Heisenberg models
are asymptotically free for z ~ 3 (Polyakov, 1975a).
Results from the Migdal recursion relation (Migdal,
1975) are reviewed and the possibility of deep relations
between two-dimensional spin systems and four-dimensional gauge theories is noted. In a short section of
concluding remarks some topics of cur rent research
interest ar e d iscussed.
The reader may wish to consult other reviews of lattice gauge theory whil. e studying this article. The review by Kadanoff (1977), which discusses fermions and
real space renormalization more thoroughly than this
articl. e, is recommended.
II. PHENOMENOLOGY
TRANSITIONS
AND PHYSICS OF PHASE
A. Facts about critical behavior
Let us begin by collecting some definitions and facts
about statistical mechanics and phase transitions. We
shall use the Ising model as an illustration. Consider
a square lattice in two dimensions with sites labeled
by a vector of integers,
John B. Kogut: Lattice gauge theory and spin systems
n=
(n„n, ) .
Place a "spin" variable s(n) at each site and suppose
that s can only be "up" (s =+1) or "down" (s = -1). Then
the energy or "Action" of the model is
S=-J
snsn+p
(2. 2)
(2.8)
Using Eq. (2.7a) X can be written in terms of a configurational ave rag e
)( = (I/Xk
T) [(s'„,) —(s„,)']
. .
[((s. . —(s. .))')],
where s„,=Q„s(n). This formula
= (1/muT)
where p denotes one of the two unit vectors of the lattice as depicted in Fig. 1, and J is positive so the Action favors aligned spins. Two important properties of
the model are (1) only nearest-neighbor
spins are
coupled. The Action is as local as possible. (2) The
If all the spins are
model. has a global symmetry.
flipped, S is left unchanged.
Placing the system of spins into an external magnetic
field B changes its Action to
S = —J
field susceptibility is a, measure of the fluctuations in
the spins. Equation (2.9) is an example of the "fluctuation —dissipation" theorem (Stanley, 1971). We learn
that X will be large at those values of the temperature
T where the spins are fluctuating considerably. X can
also be written in the form
~n
k=(1/NkT)
g (s(n)s(ns)) — P s(n))
n, m
g s(n)s(n+ p) —& g s(n) .
(2.3)
The external field breaks the system's global symmetry.
The statistical properties of the model follow from
the hypothesis that the probability for a. particular spin
configuration is proportional to
P= exp( —PS),
(2.4)
where P= 1/kT. The statistical physics can then be ob-
(2.9)
shows that the zero-
2"
which shows that X is also a measure of the correlations
among the spins. Define the spin-spin correlation function
(2. 11)
I'(n)= (s(n)s(0)),
and suppose that the system has no net magneti'zation,
(s(0)) = 0. Then, using the translational invariance of
the system,
Eq. (2. 10) becomes
tained from the partition function
Z=
g
exp(
(2. 12)
(2. 5)
-)6S),
(conf igs )
where the sum runs over al. l possible spin configurations. (On a lattice of N sites, there are 2" such configurations. ) For example, the free energy is
I = -kT lnZ.
(2. 6)
The mean magnetization
per site can be expressed eith-
er as
SS= (1/N)
g s(n))= g
(1/N)
[conf i gs)
n
(2. 10)
n
g s(n)
s
s /N
n
(2.Va)
or, using Eq. (2.6), as
(2.7b)
measure of the response of the spins to an external
infinitesimal magnetic field is given by the susceptibility per site,
A
X can diverge if the system has sufficiently
long-range correlations.
The connection between long-range correlations and
singular behavior in the system's thermodynamic properties is important. It will be discussed later in this
section and will reappear several times throughout this
article. But first we should discuss the phenomenology
of the critical region.
Consider the magnetization M as a function of T and
B. At fixed T, let B tend to zero. If M remains nonzero, the system is said to experience '-'spontaneous
This occurs only for T below a critical
magnetization.
value T, in our example, the Ising model. Otherwise
M tends to zero as the magnetic field is removed. Note
that a spontaneous magnetization indicates that the equilibrium state of the system does not possess the global
up ~ down. symmetry of the system's action. Below T,
M can serve
that symmetry is "spontaneously broken.
as a "local order parameter" to label the phases of the
system. As T approaches T, from below, M vanishes.
In many physical systems including our model it vanishes as a power
Note that
"
"
(2.13)
~
—
~ 0
FIG. 1. The unit vectors of the square lattice
sions.
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
in two dimen-
where P is the magnetization critical exponent (Fisher,
1967). It takes the value 1/8 in the two-dimensional
Ising model as noted in Table I, where other critical indices are collected. A good theory of critical phenomena should yield an estimate of P and the other indices.
Next consider the spin-spin correlation function I'(n)
for T& T, . At truly high temperature we expect thermal
fluctuations to dominate the tendency of distant spins to
align and correlate. One finds that, in fact, I'(n) falls
John B. Kogut: Lattice gauge theory and spin systems
TABLE I. Exact critical indices of the bvo-dimensional
Critical index
Ising model.
Value in 2-D Ising model
Relation
~-(T, —T)'
- lnl '" ""',(T = T.)
1/4
&(n)
'
g -(T —T, )
X-(T —T.) ~
(r —T,)-"
1.75
c-
I-a'~',
r(n)
(T = T,)
15
with the distance between spins for any
off exponentially
T above T„
- exp l:- lnI ~((»]
663
"
(2. 14)
where g(T) is the system's "correlation length.
g(T)
gives a measure of the size of patches of correlated
spins in the system. For high T, g(T), measured in
units of the lattice spacing, is of order unity. For temperatures below the critical point,
I (n) - (s(0))' Wo, ~n » 1 .
(2. 16)
There are no long-range correlations here either. The
critical temperature various thermodynamic functions devel. op singular behavior, and that this singular
behavior is related to long-range correlations and large
fluctuations. Although the underlying action of the Ising
model has only short-range forces, correlations can
appear in the model over an infinite range. One of the
aims of the first several sections of this review is to
provide an understanding of this point.
the
B. Correlation
length scaling and the droplet picture
~
system is simply magnetized.
correlation function
In fact, the connected
r, .„(n) = (s(o)s(n)) —(s(o))',
(2. 16)
is exponentially small in the Ising model. Precisely at
the critical point I'(n) falls off as a power of ~n ~:
'+"', T= T. ,
(2. 17)
r(n) —ln~
where q is another critical index recorded in Table I.
So, the system has long-range correlations only at its
critical temperature. In order that Eqs. (2.14) and
(2. 17) be compatible, it is necessary that the correlation length g(T) diverge a,s T approaches T, from above,
(2. 18)
where v is another standard critical exponent. Also,
our exercise with the "fluctuation-dissipation"
theorem
suggests that the susceptibility X diverges as T ap-
"
proaches
T„
X-(T —T, )
",
(2. 19)
Table I. Another quantity of in-
where y is recorded in
terest is the specific heat
C= —T
It also may diverge as the temperature
(2. 20)
is reduced to
Tc~
C
(T —T)
Experiment has shown that the critical indices of a
wide variety of physical systems are identical. Such
striking regularities have challenged physicists to pinpoint the essential physics of the critical region (T= T,).
One approach suggests that it is the divergence of g(T)
that is responsible for the singular dependence of all
physical quantities on T —T, (Fisher, 1967). This is
the "correlation length scaling hypothesis.
It claims
that in the critical region g(T) is the only relevant
length in the system. In some sense the system is no
longer sensitive to the lattice and its small spacing.
"
The fact that spins are correlated over distances of
order g(T) and that g(T) 1 dominates the properties of
the system.
It is interesting to accept this hypothesis and produce
a physical picture of the Ising model for T= T, (Kadanoff, 1976a). As shown in Fig. 2 we expect regions of
correlated spin of sizes ranging up to g(T). These are
depicted as droplets of overturned spins. But some
thought indicates that this picture is not quite right.
The point is that each droplet of size -g(T) is itself a
huge physical system near criticality. Therefore, to be
consistent, it must consist of droplets of overturned
spins whose sizes range from zero to g(T). A better
picture is shown in Fig. 3. According to this final view,
there are fluctuations in the system on all length scales
from zero to g(T). If we choose T= T, where g(T) diverges, then the system should appear identical on all
length scales. It would -be scale invariant.
»
( 2. 21)
where n is the specific heat critical exponent. Finally,
if we adjust T to be precisely T, and apply a small external magnetic field, we should expect the magnetization to respond sharply. A critical index 6 characterizes this effect,
M-a' ' (T=T,).
(2. 22)
The essential point of this summary is the fact that at
Rev. Mod. Phys. , Vol. 5'I, No. 4, October 1979
FIG. 2. Droplets
in the Ising model near the
critical point.
664
John B. Kogut: Lattice gauge theory and spin systems
Rules 1-3 are, in fact, not a complete list. We shall
see that universality can be discussed accurately w'ithin
the context of the renormalization group (Wilson and
Kogut, 1974). Then the origin of rules 1-3 and the additional relevant features of the physical system which
control its critical exponents can be understood. For
example,
FIG. 3. Seal. e- invariant droplets.
The correlation length scaling hypothesis leads to
lations among the various critical indices we have
3.ntl oduced.
They x'eRd
P=
2
re-
v(d-2+7)),
0=
pd~
(d+ 2 —q)/(d —2+ q) .
(2. 23)
The essential ingredient leading to these relations is the
assumption that g(T) is the only relevant length in the
problem. General relations among the thermodynamic
functions and dimensional analysis then lead to Eq.
(2. 23) (Fisher, 1967). More sophisticated arguments
based on the operator product expansion (Kadanoff,
1976a) can also be made.
It is interesting to observe that the scaling relations
are true for the critical indices of the two-dimensional
Ising model collected in Table I. It is also interesting
thai other approaches to critical behavior produce values for the indices which violate these scaling laws and
the exact results of the Ising model. For example,
mean field theory gives P=1/2, v=1/2, and @=0 independent of the dimensionality of the system. These results violate Eg. (2. 23) except in four dimensions. This
limitation of mean field theory is well known and is
easily understood by recalling that it ignores fluctuations. Therefore, although it can be a very useful
scheme, it misses the essence of the physics except in
dimension four and above. This failure is another hint
of the fact that fluctuations are an essential ingredient
in models of critical phenomena in low-dimensional,
spaces.
Since many physical systems have identical critical
indices, it is important to determine those features of
the Action which are relevant to the critical behavior of
the system. It appears that physical systems can be arranged into "universality classes, and that within each
class each substance has common critical behavior
(Kadanoff, 1976a). These classes are labled by (1) the
dimension of the local variables in the action, (2) the
symmetries of the coupling between the local variables,
(3) the dimensionality of the system. The value of this
classification is that it tells us what we can ignore. For
example, the detailed lattice features are unimportant—
they may affect the critical temperature T, but they do
not affect the singular parts of the thermodynamic functions in the critical region. For example, the Ising
model can be formulated on anisotropic lattices and the
same critical indices result.
"
Rev. Mod. Phys. , Vol.
1-3, critical
indices
pr ec is ely.
I I I. THE TRANSFER MATRIX —
FIELD THEORY AND
STATISTICAL MECHAN ICS
A. General remarks
y= v(2 —I)),
A= 2
according to rules
should not depend continuously on any parameter in the
Action. However, the low-temperature phase of the
planar Heisenberg spin system in iwo dimensions does
not have this property. Later we shall review the renormalization group analysis for that model and shall
then be able to formulate the idea, of universality more
51, No. 4, October 'f979
The goal of this section is to establish the connection
between statistical mechanics in four dimensions and
field theory in three spatial and one time dimension.
This will be done using simple examples and some general analysis. The key to this connection lies in the
transfer matrix (Schultz, 1964), as will be discussed
below. In this approach field theories are regulated by
the space-time cutoff provided by the lattice itself.
Even within this framework different cutoff procedures
can be contemplated. We shall consider both spacetime symmetric cutoffs (cubic lattices in four dimensions) a.nd very anisotropic systems in which one axis
is a continuum and the remaining three employ a lattice. The reason w'e consider these cases is that each
has certain advantages. The space-time symmetric
formulation is frequently more elegant and rigorous,
while the second approach resembles Hamiltonian quanturn mechanics and exposes some of the physical properties of.the theory quite simply.
If one's ultimate interest is field theory formulated in
a continuum space-time, then the lattice is of secondRx'y lntex'est.
It ls necessRx'y io leRx'n how' to retrieve
the physics of a continuum field theory after using the
lattice as a scaffolding on which to formulate ii precisely. We shall see that continuum limits of lattice theories exist at the critical points of the lattice theory
(Wilson and Kogut, 1974). Therefore our discussions
center upon the critical regions of the lattice theories.
In many cases we shall be interested in finding the lattice theory's phase diagram as a first step toward
understanding
its various possible continuum limits.
In. many of our discussions we shall be using different
lattice versions of a single field theory. It should be
kept in mind that these different lattice formulations describe the same physics. For example, different cutoff
procedures in perturbation theory produce identical reIt will be one of our
normalized Feynman amplitudes.
engineering problems to ensure that different lattice
formulations of a field theory lead to the same physics.
B. The path integral and transfer matrix of the simple
harmonic oscillator
Much of the formalism we wish to develop for field
theories can be illustrated in the context of nonrelativ-
665
John B. Kogut: Lattice gauge theory and spin systems
istic quantum mechanics. We shall discuss the relationship between the Feynman path integral and the
Schrodinger equation for a one-dimensional potential
problem (Creutz, 197'l). This will be worked out in detail so that later discussions of fieM theories will only
emphasize those new features encountered when dealing
with many degrees of freedom in higher dimensions.
We begin with the Lagrangian for a one-dimensional
simple harmonic osc illator,
Z = 2(x' —~'x')
.
(3 1)
In Feynman's formulation of quantum mechanics (Feynman, 1948) one considers the amplitude that the particle
will be initia. lly at (x„ t, ) and finally at (x~, t~). The amplitude for this transition'is then postulated to be
g exp[(i/n)s ],
z=
(3.2a)
where the sum is over all world lines between the initial
and final points and S is the Minkowski space Acti. on for
a particular path,
S =
Zdt.
(3.2b)
t~
So, Eq. (3.2a) states that each path contributes to the
transition through a weight exp[(i/5')
Zdt]. A sensible
definition of the "sum over all paths" must be provided
before Eq. (3.2a) becomes useful. One way to proceed
is to introduce a space-time lattice so that various
paths can be labeled simply. For example, make the
time axis discrete and call the spacing between slices c
f
(3 3)
as shown in Fig. 4. It is also best, to modify an expression such as Eq. (3.2a), which has rapidly oscillating
phases, by continuing it to imaginary time
t= —Z&
(3.4)
~
Then each path will be weighted by an ordinary damped
exponential factor and it becomes easier to distinguish
important from unimportant paths. Substituting Eq.
(3.4) into (3.2b) gives
+
(d2X2
dT
(3.5)
~
So, if we define the Euclidean Action,
S=
p
Jf
—
( )
+ tx'x*
then the path integral becomes
OQ
'
Z=
«OO
'
dx.t exp
, L. a
1
--S
For discrete time slices
&= 2
c
Q {[(x,~ - x,.)/ E] '+ (u'xg
(3.6)
.
(3.8)
Before continuing to the transfer matrix and the Schrodinger description of this system, note that Eqs. (3.'7)
and (3.8) constitute a one-dimensional
statistical mechanics problem. We have a one-dimensional lattice
whose sites are labeled with the index j. On each site
there is a variable x, which takes on values between
-~ and +~. The Action couples nearest-neighbor variables x, and x„, together. The integral J II dx, is a
sum over configurations, each weighted by an exponential exp[-(1/h)S]. It is important to note that k plays the
role of the temperature in the statistical mechanics
formulation. We usually think of the temperature as a
measure of the fluctuations in a statistical mechanics
problem and 8 as a measure of the fluctuations (through
the uncertainty principle) of a quantum problem. Recall
from quantum mechanics that the @-0 limit picks out
0 the classical
classical physics. In particular, as
trajectory of the harmonic oscillator becomes the only
path that contributes to the amplitude Z, and fluctuations
are completely suppressed. These points make the correspondence
5-
(3.9)
very appealing, since one visualizes the T= 0 point of a
statistical mechanics problem as frozen and without
fluctuations.
Now let us return to our analysis and organize the
evaluation of the integrals in Eq. (3.'l) in an enlightening fashion. Since the Action only couples nearestneighbor lattice variables x, , we can write the partition
function as
'
Z= Jt
[dx, T(x,
„,x,)],
1
(x. . -x,
T[x„„x,) xxpI- ——
.
I
(3.10)
.
where
1
dx,
(3. I)
)*.
=
+ —co 2 gx 2k+1 + —40 2gx.2
jJ
2
2
(3. 11)
~
We can think of T as the matrix element of an operator the transfer matrix and establish the equivalence
of the path integral with the Hamiltonian approach to
quantum mechanics. First we must set up a space of
states. Let there be operators x and p. Eigenstates of
x are states in which the particle is localized at the
—
ted+i
/j
l
I
I
I
—
position x,
l
I
X X
I
Xg
X)+I
discrete time axis and
Xb
XI
FIG. 4.
A
il. lustrative
mechanics.
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
path in quantum
X X
(3.12)
~
The states ~x) have the continuum
normalization
(3.13a)
666
John B. Kogut: Lattice gauge theory and spin systems
We assume that they are complete,
so
(s. 1sb)
Finally, let there be a momentum
ly conjugate to z,
[p, x] = -ie.
Using the operators p and
erator T with the property
~i~x&=
&x
operator p canonical-
(3.14)
g we shall construct an op-
H = —(h/c) lnT,
(3.20)
is rather complicated and not very enlightening.
(The subscript s anticipates useful notation for later
discussions. ) However, if we let c-0 so that the T axis
becomes a continuum and there are an infinite number
of slices between v, and 7» then
but it
th
T(x, x),
(3.15)
c.
time) evaluated over the interval
tain a useful expression for Z,
Using T we can ob-
T"
~
(3.21a)
H,
where
by
d, &x,.„~
Formally one can consider the
operator
H
Eq. (3.11). Clearly T is the usual time
evolution operator of quantum mechanics (imaginary
as given
Hamiltonian.
quantum
H=
A
'(p'+ ~'x').
(S.21b)
—,
H is the familiar Hamiltonian for the simple harmonic
oscillator. The reason H is simple while H, is not is
the following. To write Eq. (3.19) as a simple exponential we apply the identity
~
exp(A) exp(B)= exp(A+B+
&x~
~ ~ ~
T~x„, &dx
~
))&, &x&)&,
~
T~
x))-&2&dx&(-
2&x
~
(3.16)
where N —1 is the number of 7 slices between T, and 7~
and we used completeness in arriving at Eq. (3.16). If
we impose periodic boundary conditions and sum over
all possible initial positions of the particle, Eq. (3.16)
is replaced by the more familiar result,
(3.17)
Z trTN
So, if we can find an expression for T we are done.
'
~
exp( ——,cp')
~x&
itesimal step in time (Messiah, 1962). The Schrodinger
equation follows by observing that the Feynman interpretation of Z means that if the wave function of the
particle is P(x, r), then
x')=
f
i1-(
~
f
d)
d&'&'I&'&«'I
1
Then Eq.
t dpdp' exp(ip'x'/k)
h
x exp(--,'cp') 6(p' —p) exp( ipx/5)
1
(2m)
~
dp= exp[ip(x' —x)/5]
' cp')
x exp( ——,
= const. exp [—(2ch ')(x' —x)'] .
Using this result and inspecting
Eq. (3.11), we have
exp [—(1/25)cp']
and this is true for all lattice spacings
ordering in Eq. (3.19) is important.
c.
The operator
The transfer matrix is closely related to the theory's
51, No. 4, October 1979
so that this inter-
&
6(x'-x) -(c/e)H6(x'-x).
(3.25)
(3.24) becomes
9
—
g(x,
T) =
—H(t&(x, T),
(3.26)
(3.27)
which is the Euclidean version of Schrodinger s equation. H in Eq. (3.27) is just the Haniiltoman in the xspace realization. In summary, we have the usual operator form of quantum mechanics in the Schrodinger
picture operators are 7 independent and state functions
carry their 7 dependence.
Since the path integral and Hamiltonian formulations
of quantum mechanics are equivalent, convenience dictates which one uses in a certain application. Generally, the path integral is better for scattering processes,
while the Hamiltonian is better for bound-state problems.
(3.19)
x exp[-(1/4h')c&u'x'],
Rev. Mod. Phys. , Vol.
/a)Hi
(3.24)
—
(3.18b)
[- (1/45') c&u'x']
=
y(x', ~') = y(x, ~) -(c/m)Hy(x', ~)
(2m)
T= exp
Z(x', x';x, x)p(x, x)dx.
Choosing 7 —T = c to be infinitesimal
val consists of just one slice,
= const. exp[ —2 c@'(x' —x)'],
'&I*&=
(3.22)
which we identify as the familiar definition of the Hamiltonian in terms of the evolution operator for an infin-
(S.16a)
which is easily obtained using the complete set of momentum eigenstates f p&},
&x'l«p&--")
. ),
B= —c~'x'/4k'. But [A, B] =O(c'),
so as c, -O it is negligible compared to A and B. Therefore
T= exp{—c/h [H+ O(c)]},
(3.23)
&t(x',
Note that
&x'
[A, B] q
with A= —cp'/2h and
&x, T ~x, &dx, &x, T ~x.&
~
2
C. The transfer matrix for field theories
Now we shall extend the
section to field theories.
considerations of the previous
Most of our efforts will be
John B. Kogut: Lattice gauge theory and spin systems
a„f(n) = f(n+
aimed at establishing precise correspondences between
properties of a statistical mechanics system and those
of a field theory.
Consider a self-coupled scalar fiel. d in d space —time
dimensions. The Lagrangian will be
d(4 1)+
.
2
Qt 2
Q
2
2
Q
2 p
2
K =(1/2~)a'
50= p 7~ d-1 po2
(3.28)
P{(l/sa)a" [a„b(n)]*a-'. na'
+-,'
'
g [ab(
where n= (no,
n„n„n, )
s=
Q
~
'K
J~, [@'(n) —@(n)]'+ —,
'ra'
K= —
Qo=
~
'
7~d-1 Ao.
(3.31)
K( a.
S)'+Kg (a,b)'+b, b'+nb'j.
g {[p'(n+
k) —@'(n)]'+
[p(n+ k) —g(n)]'}
'
(3.34)
~
To obtain an operator expression for T we introduce second-quantized
variables,
[f(n'), @(n) j = ib,
Then manipulations
fields (t)(n) and
Pr(n), which
-
generalizing
a
four-dimensional statistical mechanics problem. The
bounda, ry conditions of Eq. (3.33) might consist of specifying @ on an "initial" temporal slice and a "final"
temporal slice in direct correspondence with the example of the previous section. Again we can introduce
a transfer matrix to "propagate" the field (t)(n) in the
temporal direction. We label the field (t) on one time
slice and P' on the next time slice, and define
lat-
' b, "(n)+ @'(n)]+-, M, [(t "(n)+ y'(n)]
+ —,
[y
(3.32)
(3.33)
~t dy(n}e
which can be thought of as the partition function of
tice, a„ is a discrete difference operator
exp —
g{
Z=
(3.29)
labels the four-dimensional
'
The path integral for the lattice theory is
)]*
aa" a,*b'(n)+ aa "b,b'(n) j,
(3.30)
(k=1, 2, 3) la-
Then K, is the strength of the nearest-neighbor coupling
in the temporal direction and K is the coupling in the
spatial direction. Note that they are equal only if v = z.
Equation (3.29) is now
and a formal expression for the path integral can be
written down. As in the previous section it is best to
analytically continue these expressions to imaginary
time. We assume that this can always be done. It can
be done order by order in perturbation theory for all
the Green's functions of the theory (Schwinger, 1958).
Granting this, we formulate the theory on an anisotropic
space-time lattice by replacing integrals by sums and
derivatives with discrete differences. Denote the lattice spacing in a "spatial" direction by "~"and that in
the "temporal" direction by T. Then the theory's Euclidean Action, the temporal integral of the Lagrangian,
becomes
s=
-f(n},
p= 0 is the temporal direction, and p= k
bels the spatial directions. The notation n= (n o, n) will
also be used from time to time. It is best to name the
coef f ic ients in Eq. (3.29)
4
g
p)
are conjugate
(3.35)
those of our previous section give the partition function
Z= trT~"
(3.36)
where N is the number of time slices (spacing 7') between the initial and final times. The trace appears here because we have identified the initial and final fields and have summed over it. The operator expression for T is
I
S' enp
—,
.
+{Kg [4'(n+b)
g (1/4K,
x exp —
)Pr'(n)
—b'(n)]*nb S (n)+n
exp —2
g
Kg
S'(n')j
[P(n+ k) —P(n)]'+ b, g'(n)+u,
In general T is not the exponential of a simple, familiar
operator. If we choose r = a so that the lattice is symmetric, we can still write
A
A
e H~V'
(3.38)
if we are willing to consider a complicated "HamiltonRev. Mod. Phys. , Vol.
51, No. 4, October 1979
@
(n)
(3.3V)
ian" H . The notion of infinitesimal time translations
does not exist here, but the statistical mechanics problem is quite elegant because K= K,. Alternatively we
can take the v-continuum limit and compute the same
partition function by subdividing the overall time interval into an infinite number of steps. In that case v -0
S)bb,
668
John B. Kogut: Lattice gauge theory and spin systems
while a is held fixed and
T= e H'=1 —rH+ '
1S
''
(3.39)
where H is the familiar canonical Hamiltonian of the
original field theory [Eq. (3.28)] formulated on
a spatial lattice. From Eq. (3.31) we note that, as r
for fixed a,
K, -~
K-0
and
—0
(3.4Oa)
in such a way that their product remains
2(g- 2)
fixed,
(3.40b)
This result is not surprising: in order that the physics
be the same in various lattice formulations of a theory,
the couplings must be adjusted appropriately.
We see
that taking a r-continuum limit forces us to consider
very anisotropic statistical mechanics systems.
Now we want to establish detailed connections between
the partition function and the operator formulations of
a field theory. We shall do this within the context of
the symmetric formulation v = a for ease of presentation. The 7-continuum formulation could also be used;
the details are left to the reader and later sections of
this review. So, we have a statistical mechanics problem on a symmetric lattice and field-theoretic formulation using operators Q(n), f(n), and H, defined on a symmetric lattice which provides the theory with an ultraviolet cutoff. The correspondences we shall find are:
Statistical Mechanics
Eield Theory
Free energy density
Vacuum energy density
C orrelation
P ropagator
function
(3.41)
To obtain the partition function we need T raised to a
large power N+ 1,
i) exp [—(K+ 1)E,7 ] &i .
l
(3.42)
Now a significant simplification occurs as N- ~. Suppose that the lower eigenvalue of H, is unique, S'p. Then
the first term in Eq. (3.42) dominates and the relative
error in dropping all other terms goes to zero exponentially as N-,
o&e
so
F=fVT,
(3.46)
f
where V is the volume of space and
is a free energy
density. Similarly Ep is an extensive quantity in three
dimensions, so
Ep = upV,
(3.47)
where (d, is the energy density of the quantum
Collecting these results,
vacuum.
(3.48)
as claimed above. The reader should note that the free
not the internal energy —
of statistical mechanenergy —
ics enters this correspondence. E contains entropy effects,
F= U —TS (T= temperature),
(3.49)
where U is "internal" energy.
The assumption that the largest eigenvalue of the
transfer matrix be unique is an important one. We shall
see that it means that the system is in its high-temperature phase.
propagator
in Min-
where 0) is the exact ground state of H, and ()))(x, t) is
the field operator expressed in the Heisenberg picture.
@(w, t) is related to the Schrodinger (time-independent)
fields we have been using by
l
ie
l
s1onsq
&(t»= &0ITi(t )i«»lo)
Mass gap
These results will. give us additional insight into the nature of phase transitions and the continuum limits of
lattice field theories.
A
Since the transfer matrix is Hermitian (T= T~), we
know that its spectrum can be arranged into an orthonormal set (l 1)) with real eigenvalues [exp(-E,.v )]. A
spectral decomposition can be written
T""= Q
where we have absorbed a factor of kT into our definition of F. But F is an extensive quantity in four dimen-
Next consider the field-theoretic
kowski space
Reciprocal of the
correlation length
T=
(3.45)
"&o
(3.43)
where T is the difference between the initial and final
times. So, the partition function becomes
Z e AT
(3.44)
'"".
y(t, x) = e'""y(x)e
Choosing t&0, Eq. (3.50) becomes
t), (t, X) = &0 @(X)e '" ' td(0())l 0)e'eo',
(3.52)
l
where x and t label points on the space-time lattice.
Now
consider the correlation function,
I'(n„, n) = Z
'
f
dd(n, ', n')d(n„, n)d(0, 0)n * . (3. 58)
'
n', a'
Organize the configurational sums so that for each
time slice np' the integrals are done over all spatial
sites n. Then, only for the slices n,'=0 and np np will
the calculations be different from those for Z itself. So,
we find a factor of &@'l Tl(t)) appearing between adjacent slices. Following the steps which lead to Eq.
(3.36), it is easy to obtain an operator expression for
the correlation function,
"
1(n„n) = tr{T @(n)T"0@(0)T )/trT
(3.54)
where P+ np+ I = N+ 1. Now let the number of sl. ices I',
n„and L go to infinity so that Eq. (3.43) can be used
for the transfer matrix. We then find
n
But the free energy of a statistical mechanics problem
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
I'(n„n) = 0
&
l
@(n)e Oe~'p(0) 0)e ohio'.
l
(3.55)
669
John B. Kogut: Lattice gauge theory and spin systems
this result with Eci. (3.52) we have the iden-
Comparing
ti
A. Transfer matrix and r-continuum
I'(n, . n) = h(-in, T, n),
(3.56)
which gives the second correspondence of interest. So,
by computing the statistical mechanics correlation function, we learn the propagator of the field theory at
imaginary time. The analytic continuation to real time
must then be done. The general postulates of field theory (Streater and Wightman, 1964) guarantee that propagators have sufficiently simple analytic structure that
this continuation is possible.
From these results it is easy to relate the mass gap
m of the field theory to the correlation length g of the
statistical mechanics. This important result will be
m= 1/(ga) .
nentially,
I(. 0)- xp(- .I&(),
,
(3.58)
I
»
in7', 0).
g. Now consider the propagator b( —
Inol
Inserting complete sets of states (I l)] of the Hamiltonian H, into Eq. (3.52), we find
A( —
in~,
0)=
P
exp[ —(E, Eo)~&] (01 '(
I
)
I&)l'. (3.59)
I
For
1,
the right-hand side of this equation will be
state having the smallest value of E',
—E,. But this is just the lightest pa. rticle state at zero
momentum.
The matrix element (0l@(0) Il) is almost
certainly nonzero in this @' field theory, and the smallest value of E, —E', is m, the physical mass of that par@gal»
dominated
by that
ticle. So,
&( —
in', 0) —exp( —mnr) .
(3.60)
Using the general relation Eq. (3.56) and comparing Eq.
(3.58) with (3.60), we obtain our desired result, Eq.
(3.57). Note that 7 = ~ here because we are using a sym-
metric space-time lattice.
Now we can return to our assumption that the largest
eigenva, lue of the transfer matrix is unique. This means
that m must be different from zero, which in turn implies that g must be finite. In other words, the parameters in the Action must be chosen so that the system is
not critical. In addition, the temperature of the underlying statistical mechanics must be above T, . To
understand this consider the Ising model. Below T, the
system is magnetized and the two spin configurations,
one with magnetization M and the other with magnetization -M, have identical free energies. The largest
eigenvalue of the transfer matrix is then doubly degenerate for T&T,.
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
formulation
In this section we shall begin to illustrate the general
remarks of the previous section. %e shall emphasize
the v -continuum Hamiltonian approach to the two-dimensional Ising model (Fradkin and Susskind, 1978) and
shall discuss partition function analyses of the model
formulated on a symmetric lattice in a later section.
Consider a two-dimensional lattice and place variables o', (n) =+1 on sites. Denote the unit lattice vector
in the temporal direction by v' and that in the spatial direction by z. Then the Action is
P [P,o,(~+
S= —
(3.57)
This formula implies that to make a field theory with
particles of small masses, the underlying statistical
mechanics must be nearly critical. In particular, taking the continuum limit ~-0 we shall reach an interesting theory, i. e. , one in which there is an excitation
spectrum of finite masses, only if $ —~. One of our
main interests will be, therefore, the character of lattice theory's phase diagrams and the nature of their
critical regions.
To establish Eq. (3.57) we recall that the correlation
function of a system not at a critical point falls expo-
for
lSING MODEL
IV. THE TWO-DIMENSIONAL
&)o', (~)+ Po.(~+ x)o.{n)
],
{4.1)
where the temporal coupling P, and the spatial coupling
Let us construct the transfer
P are free parameters.
matrix and find the 7-continuum Hamiltonian of this
model. It is then better to write the Action as
S= 2 P
g [o3(n+ 0) —
o'~(n)
] —P
g
o'3(n+
x) cr~(n), (4. 2)
which differs from Eg. (4. 1) by an unimportant constant.
Consider two neighboring spatial rows and label the
spin variables in one row o,(m) and those in the next
row s, (m). The argument m runs over the integers la. —
beling the sites of a spatial row as shown in Fig. 5. The
Action can now be written as a sum over these rows
S=g
L(n, +1, n. ),
(4. 3a)
no
where
L= —,' P, Q [s,(m) —(r,(m) j'
—2
P
Q
[v,(m+ 1)o,(m) + s, (m+
l)s, (m)] .
(4. 3b)
If there are M sites on each spatial row, there will be
2~ spin configurations on each row. Therefore the
transfer matrix will be a 2~ x 2 matrix. Consider a
diagonal element of the matrix. Then a, (m) = s, (m) for
all m, and Eg. (4.36) reduces to
L(0 flips) = —P
g v, (m+ 1)o,(m) .
(4.4)
If there is one spin flipped between the two rows, then
L(1 flip)
= 2P
—2 P
Q [o3(m+ 1)0~(m)
+ s, (m+
1)s,(m) ],
(4. 5)
Sp(Al)
spoce
cr5 (m)
I'IG. P. Spin variables on adjacent spatial rows of the twod imens iona l Is ing model.
670
John B. Kogut: Lattice gauge theory and spin systems
and if there were
T(1 flip}=T= —IH
spin flips,
yg
L(n flips)= 2nP, —2 P
P [(I,(m+ 1)(I,(m)
+ s, (m+
1)s,(m)] .
(4. 6)
Next we must determine how P and P must be adjusted
so that the trans f er matr ix has the f orm
T= e 'H=1 —7.a
~ ss
(4.7)
as v, the lattice spacing in the time direction, approaches zero. Consider various matrix elements of T,
pp s (I+1)s (m)I
r(p flips)= sxpI
m
rh
1
T(l
TH~
(4.8a)
III
flip) = exp( —2P, )
x exp
(4. 12b)
(4. 12c)
T (n flips) = T = —TH
Equaf:ion (4. 12c) implies that the only matrix elements
of II which survive as ~
are the zero and one-spinflip cases. This is a nice simplification which only occurs in the limit. To write H in operator form we need
spin-flip operators. Place a Pauli matrix g, on each
-0
site
nz,
(4. 13a)
Let spin up (at site m) be represented by the vector (', )
and spin down by (0). Then (1,(m) is a spin-flip operator
for site nz. To check this recall that in the representation where Eq. (4. 13a) holds one also has
(4. 13b)
-,'-
0-,
m+ 1 o,
I
+ s,
~+ 1 s, m
so
m
(4. 13c)
(4.8b)
a. s
claimed.
To produce Eq. (4. 12) the Hamiltonian
must
be
T{n flips) = exp( —2nP, )
X exp
a= —
o, Pl+1
2
03 Rl + s,
kB+1
s,
(4. Bc}
These equations will determine the 7 dependence of P,
and P. From Eq. (4. 8a) we learn that
(4.8b),
exp( —2P, ) -
v.
.
(4.9b)
Therefore P and exp(-2P, ) must be proportional.
constant
the proportionality
P = X exp(
~ = exp(
Define
X
—2P, ) .
(4. 10)
We can identify the temporal
lattice spacing
—2P, ),
so that the coupling between nearest-neighbor
a spatial row is
spins
within
detailed scaling relations are different here. The physical interpretation of the parameter A. and the scaling
relations will. become clear as we continue.
Using the relations Eq. (4. 11), we can identify the
Hamiltonian H. Equations (4.8) become
T(0 flips)=1+
7
XQ 0,(m+1)o, (m) =1 —TH],
m
„„.
,
(4. 12R)
Rev. IVlod. Phys. , Vol.
51, No. 4, October 1979
m+ 1 0, m
.
(4. 14)
So, we have a one-dimensional quantum Hamiltonian
which can be interpreted as an Ising model in a transverse magnetic field (Pfeuty, 1970). We shall use it to
study the phase diagram and critical region of the
Ising model. According to
our general considerations it must have the same phase
diagram and critical, indices as that model.
It is interesting to understand Eqs. (4. 9)-(4.11) in
more detail. Consider the formulation of the Ising model, Eq. (4.1), which began this discussion. That Action
is parametrized by two variables (P„P). It is known
that for a certain range of these variables the system
is magnetized (ordered, ferromagnetic), while for other
VRhles lt ls Ilot. 111Rglletlzed (dlsol del ed, pR1'R111Rglletlc).
There is a critical curve separating these two phases in
the (P„P) plane. In a later section, in which we shall
study Eq. (4. 1) and its partition function directly, we
shall discuss calculational methods which easily yield
the critical. curve
sinh(2P, ) sinh(2P) = 1.
(4. 11b)
In summary, these results show that in order to define a smooth 7-continuum theory the couplings must be
adjusted so that the temporal coupling grows large while
the spatial coupling becomes weak. This general feature was noticed earlier for the (t) field theory, but the
I
cr,
"classical" two-dimensional
(4. 9R.)
and from Eq.
—x
Rl
„„,,.
~H/
(x, rn
{4.15)
This result is drawn in Fig. 6 and the phases are labeled appropriately.
Choosing ( P„P) on the critical
curve produces a theory with an infinite correlation
length. Consider the form of the critical curve in the
limit P, —. Then
sinh2P,
——,' exp(2P, )
(4. 16a)
and
sinh2P- 2P.
(4. 16b)
So, to maintain criticality in this limit, the parameters
must satisfy
P= exp{-2P,).
(4. 17)
671
John B. Kogut: Lattice gauge theory and spin systems
FIG. 8.
ical curve
A
curve of constant
I'(n) if p~ &p.
FIG. 6. Critical curve and phase diagram of the anisotropic
tv o-d ime ns ional Is ing model.
/
Note that this is the same scaling relation we found in
Eq. (4. 10) except that X has the specific value of unity.
This shows that we can view the T-continuum version of
the theory as a natural limiting case of the general
model. , and that the parameter X can be used to label
its phases. If X& 1, the model lies in the disordered
phase. So A is a temperaturelike variable with the cor-
respondence
I/X
- temperature .
(4. 18)
refer to 1/X simply as temperature.
critical temperature is A. = 1.
We shall often
The
It is interesting to understand the scaling relation Eq.
(4. 10) more intuitively (Fradkin and Susskind, 1978).
To do this consider the spin-spin correlation function
for the general model Eq. (4.1). If P, = P, the lattice is
symmetric under rotations through 90', and the correlation function I"(n) = (o,(n)o, (0)) shares this symmetry.
Curves of constant I'(n) are shown in Fig. t for large
They are approximately circular. If P, is now increased, correlations in the v direction will become
special significance of the point
called a "duality" transformation
1. This mapping is
(Kramers and
X=
Wannier, 1941).
C onsider the one-dimens ional quantum-mechanical
formulation of the model Eq. (4. 14). From here on we
shall omit the "hats" from operators. The first element of the duality transformation associates a new
("dual" ) lattice with our original spatial lattice. Sites
of the original lattice will be associated with links of
the "dual" lattice and vice versa. The transformation
is visualized in Fig. 9. Operators are placed on the
dual lattice so a spin system complete with a Hamiltonian will be generated from the original system. Define
the operators on the dual lattice
p, (n) = o, (n+ 1)a,(n),
V.(n) =
.
a, (m)
(4. 19)
m&n
So, p, (n) senses whether the spins on adjacent sites are
aligned or not. p, (n) flips all the spins to the left of n
The dual operators have several important properties:
(1) They satisfy the same Pauli spin algebra as a, and
a, . (2) The Hamiltonian can be rewritten simply using
the dual spin operators. To check the first point, recall the algebra that defines the Pauli matrices. On a
stronger, so curves of constant 1" will be distorted into
ellipses with their major axes in the 7. direction. An
example is shown in Fig. 8. However, suppose we demand the same physics of the two lattice formulations.
Then we must compensate the inequality P, & P by distorting the lattice so that T &~. If the lattice is adjusted
appropriately, the contours of constant I' can be left
invariant. Clearly, as P, —~, the lattice spacing in the
time direction must be squeezed to zero. The result
r = exp(-2P, ) is the sealing relation which insures that
the long-distance physics of the various formulations is
kept essentially unchanged.
and on different
B. Self-duality of the Ising model
Our first step towards developing a detailed under-
One checks that p~(n) and p, (m) satisfy the same
bra by using Eqs. (4. 19) and (4. 20) directly. For
standing of the Ising model is to derive a mapping between its high- and low-temperature behaviors. As a
by-product of this construction we shall understand the
constant
I'
given site,
a, (n)o, (n) = -a, (n)a, (n),
a,'(n) = a,'(n) = 1,
(4. 20a)
sites they commute,
[a (n}, a (m}] = 0, etc. , if n Wm.
ample, to check that
(4. 21)
p, (n) p, ,(n) = -p, (n) p, (n),
note that when Eq. (4. 21) is written out only one factor
of o, and g, are on the same site. Then the first relam m+
I
Duality
FIG. 7.
A curve
of constant I'(n) for the symmetric
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
model.
(4. 20b)
algeex-
FIG. 9. Dual transformation
for a spatial. lattice.
672
John
8. Kogut:
Lattice gauge theory and spin systems
tion in Eq. (4. 20a) implies Eq. (4. 21). One proves that
(4. 22)
p, (n)p, ,(m)= p, ,(m)p, (n), n&m,
's
's
by observing that an even number of 0, and 0, are
interchanged when passing from one side of Eq. (4. 22) to
the other. An even number of interchanges always produces a positive sign.
The second point is verified just as directly.
Since
(4. 23)
the Hamiltonian
is
All of these approaches will be used within this review.
But first we shall consider method 3 and discuss strong
and weak coupling expansions for the quantum Hamiltonian formulation of the Ising model. From these calculations we shall determine the critical temperature
A. , the mass gap critical index p, and the magnetization
index P. Because of the simplicity of the model, exact
answers will be found in al. l cases. It is probabl. y
worthwhile to understand these expansion methods since
they apply equally simply to more complicated theories
where exact solutions and renormalization groups are
not available.
We begin with a calculation of the mass gap of the
Hamiltonian,
(4. 24)
H=
n
n
which has the same form as the Hamiltonian
terms of the 0's,
written in
[1 —o;(n)] —&
Q cr, (n)o, (n + 1) .
Consider the theory at high temperature,
small. If we write
') .
(4. 25)
's
But since the o's and p, have the same algebra, this is
really an expression of symmetry for the original modit state's that the high-temperature and lowel alone —
temperature properties of the model map onto one
another! It is called "self-duality. " In particular, Eq.
(4. 25) implies that each eigenvalue of H satisfies the
H(a". X) = XH( p. ; X
Q
(4. 29)
i. e. ,
A.
very
H= Ho+ XV,
(4. 30a)
with
(4. 30b)
relation
(4. 26)
This has a very important implication. Consider the
mass gap G(&) of the Hamiltonian as a function of X. Suppose it vanishes at some particular point. This would be a
critical point of the theory because the correlation
length would diverge there. But Eq. (4. 26) states that if
the gap vanishes at a certain A. , it must also vanish at
Therefore, if we assume that the critical point is
unique, the self-duality of the model implies that it occurs at
(4. 2'I )
This result agrees with our earlier observations. In
the next section we shall compute the mass gap and find
(4. 28)
in agreement
with these considerations.
Note that the self-duality of the model yields the critical. point only if one assumes that the point is unique.
This is a rea. sonable (and true) assumption for the simple Ising model, but it is not true for more intricate,
self-dual theories (Elitzur et ~l. , 1979).
then we are ready to do perturbation expansions in A. .
First we must determine the X=0 ground state. To
minimize H, we must choose the spins at every site
such that
(1) Solve the model exactly.
(2) Renormalization group calculations.
(3) P erturbation expansions.
Rev. Mod. Phys. , Vol.
51, No. 4, October 'l979
I
I
I
To calculate the mass gap we must obtain the expansion for the ground-state energy and the first excited
state. In the X= 0 limit, the first excited state above
IO) consists of just one flipped spin. This state is N
fold degenerate (& is the number of sites of the lattice)
because the flipped spin could occur anywhere on the
spatial. lattice. However, this degeneracy is resolved
by constructing states which have definite momentum.
Consider the zero-momentum state
C. Strong coupling expansions for the mass gap, weak
coupling expansions for the magnetization
How does one determine the phase diagram and compute critical indices for a statistical mechanics system?
There are various methods:
0) = 0) (for all n) .
(4. 31)
It is convenient to refer to 0) as having all spins "up. "
This terminology is different from that used earlier in
connection with Eq. (4. 13). In that basis the eigenstates
of o, are (', ) with eigenvalue +1 and (', ) with eigenvalue
-1. Then 0, acts as a spin-flip operator within this
basis. Therefore we can describe the perturbation V,
Eq. (4.30b), as an operator that flips the spins on adjacent sites.
o', (n)
1)=(1jWpr)
g ~,
(4. 32)
(&) Io),
—1) inwhose energy is the mass gap. The notation
dicates that 0, has flipped one spin down. The state
equation (4. 32) is properly normalized, i. e. , (-1 -1)
I
=
1.
I
We want to calculate the X expansion for the mass gap
to sufficiently high orders that we can sensibly discuss
John
8. Kogut:
673
Lattice gauge theory and spin systems
X near unity where the phase transition is expected. So, we need the formulas of Raleigh-Schrodinger perturbation theory to high order. They are
easily obtained from the elegant signer-Brillouin formulas (Baym, 1969}. The expansion for the energy of
a state la) whose zeroth-order energy is c,(H, ja)
~o 0
(4. 33a)
E, = c, + c, A. + c,,X'+ c,X'+ c X +
values of
FIG. 11. Two possible actions of the potential.
where
initially containing
c, = (al Vla),
c, = (a Vg Vla),
B., = (a vgvgvla) —(a vja) (al vg'via),
~, = &al vgvgvgvla) —&al vgvla) &al vg'via)
l
l
l
l
l
(4. 33b)
l
and
g is
the resolvent
}.
g= (1 —la) (a }/(~, -H,
(4.33c}
It is easy to apply these formulas and determine the
mass gap. It is helpful. to represent the terms of the
perturbation expansion pictorially. Let a single vertical
line (Fig. 10) represent a flipped spin at a certain site.
The potential V can act on the flipped spin and move it
one lattice spacing to the left or right. Alternatively,
V can act on the X = 0 vacuum and flip two nearestneighbor spins over. These effects are shown in Fig.
11. These figures should be read vertically as a guide
to Eq. (4. 33b}. For example, Fig. 11(a) shows a contribution to (al Vla): the'flipped spin is initially on site
n, then V, which is represented by a horizontal link,
acts and flips that spin down while raising a nearest
neighbor. The spin-up nearest neighbor is present in
the final state. Figure 11(c) shows a spin configuration
which contributes to the state Vja).
I et us compute the gap through X'. The zeroth-order
energy is just co= 2. This is so because when one spin
is flipped, one term in 0, is increased by two units.
The first-order coefficient c, is -2. It is minus because of the minus sign in the expression for V, and
the 2 records the presence of iwo graphs, Fig. 11(a) and
11(b}. To obtain this result mechanically, evaluate the
l
matrix element
&-1l Vl-1)=
-(1~~) Q
g o,(m)o, (m+ 1)o'3(n')
There are two contributions:
= m and rn+ 1= n. Since 0,'=
n= m and
l
0).
n
n+
Fig. (2(a)=2(
)3*,
Fig. 22(b) = (N —2) (
}3'.
The third-order contributions are obtained similarly.
There are contributions from the (al VgVgVla) piece of
E(l. (4. 33b) shown in Fig. 13, as well as the —(al Vla)
(a Vg Vla) piece which is easily obtained from the
lower-order calculations. Collecting these results
l
Fig. (3(a)= —2(
Fig. 13(b)= —2(
}3',
g)l',
Fig. 23(c)= —2( d)1*,
Fig. (3 (d) = —2(N —3) ( 3 3) 3
-(al via)(al vg'via)=
-(-»)
16+
(~-2}
m+1=n' or n'
1, each term gives a fac-
n
.
(b)
to the energy of ) —1).
The dashed line denotes an intermediate state.
FIG. 12. Second-order contributions
I
Rev. Mod. Phys. , Vol.
(4. 35)
(4. 34)
FIG. 10.A flipped spin on site
n-I
tor N, canceling the normalization N '. Other terms in
the sums contribute nothing, since in those cases the
state o,(n)a, (m}cr,(m+ 1)o,(n') 0) contains some flipped
spins, so its projection onto l0) vanishes.
Second-order graphs are shown in Fig. 12. In these
cases there is a sum over intermediate states. The
energy denominators are (2-6) ', since 2 is the energy
of the initial state and 6 is the energy of the intermediate state (three spin flips, ea, ch producing two units of
energy). The intermediate state is labeled by the
dashed horizontal line in the figure. The second-order
results organized by figure number are
To finally obtain the mass gap we must calculate the
shift in the vacuum energy through this order in A. and
take the difference of the two series. The zeroth-order
&0jo3(n)
n, n'
x
state
l
+ (a lV la) (a Vla) (a Vg'Vla)
—(a Vla) (al Vgvg V+ Vg'VgVla)
V on a
a flipped spin.
51, No. 4, October 1979
John B. Kogut: Lattice gauge theory and spin systems
the resulting equation (4.39), which was derived only
for A. &1, can be extended to all ~,
(4.42)
as was recorded earlier.
.
In this case we
organize the calculation around zero temperature,
1/A. =0, where the system is definitely magnetized,
and
develop a perturbation series in powers of 1/A. . To
begin recall a simple fact about perturba, tion theory.
Suppose we want to calculate the matrix element of an
operator 6 perturbatively.
Then if we modify the
Ham iltonian
Now we turn to the magnetization.
(b)
(4.43a)
where h is a parameter, the matrix element of
state lit», an eigenstate of H with energy E, is
in the
(9
(4.43b)
&E(I )
(c)
FIG. 13. Third order e-ontributione to the energy of —i&.
dashed lines labe1ing intermediate states are shown only for
the first graph.
I
—the
constant term Q„' 1 was added to
to achieve this. The first-order correction is also zero, since Vl0& does not project onto
energy is zero
the Hamiltonian
Second-order effects are shown in Fig. 14,
~0&.
'.
Fig. (14) =-(N/4)A.
Third-order effects vanish identically.
This equation is just the familiar statement of perturba, tion theory in the term h6l about the Hamiltonian H. The
useful feature of the formula is that in some applications it is convenient to calculate E(h) directly and take
its first derivative at h = 0 afterwards.
To develop perturbation series about 1/X =0, consider
the operator
g a,
(1/A. )H = —
(n)o', (n+ 1) —(1/A, )
(4.37)
Collecting these results we find the series for the gap
G (A. ) = (2 —2)A. + (0)A.' + (0)A.' +
~ ~ ~
where the second- and third-order coefficients have
vanished identicallyl
Higher-order calculations confirm the obvious suspicion
the series truncates after
the first term. So, we find an exget result
W=
&1. Note that
at , =1. This proves
vanishes
that the Ising model has just two phases separated by a
continuous phase transition. We also learn from Eq.
(4.39) that the critical exponent v is unity
G(A. )
.
(4.44)
g [1 — (n)o, (n+1)] — g v, (n).
(1/A. )
&,
(4.45)
Finally, choose the operator
(4.46)
and incorporate
(4.39}
A.
&, (n)
It is more convenient to add constants and use the
operator 8'
—
for
g
n
n
W- W+h
A.
it into W,
g o, (n).
To do perturbation
(4.47)
theory define the operators 8'0 and
V
W = Wo + (1/A. )V,
(4.48a)
(4.40)
which is another exact result. The fact that we have
obtained the correct answer (see Table 1) constitutes
an example of universality
the singularities in the
critical region of the model are independent of its
deta, iled l. attice structure.
Since the gap must satisfy the duality condition Eq.
—
(4. 26)
(4.41)
(4.48b)
To begin we must determine the ground state of Wo.
It would be doubly degenerate, having either al. l spins
up~
o (&)lo& = lo&
or all spins
FIG. 14. Second-order contribution to the vacuum energy.
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
(all &),
"down, "
(4.49a)
o, (n)l0& =-l0& (all n),
(4.49b)
if h were set to zero (The ter.minology for "up" and
John B. Kogut: Lattice gaoge theory and spin systems
(,
( 4
Fig. 16(b) =
FIG. 15. Second-order contribution
ground-state energy.
to the 1 jX expansion of the
"down" now refers to the eigenstates of O', . The operator 0, acts as a spin-flip operator, as in the discussion
of Eq. (4. 13}.} However, the operator hg„a3(n) breaks
the global up
down symmetry and picks out one of the
two degenerate ground states. Choosing k to be small
and negative singles out the a, ll. -spins-"up" state as the
true ground state around which we shall do perturbation
theory. The perturbation V now fl.ips spins in this state.
As we calculate to higher and higher orders in 1/A. we
shall encounter intermediate states with more and more
spins turned over.
Since we want the ground-state expectation value of
the magnetization, we must calculate Eo(h}, the energy
of that state. In zeroth order we have hN, where N is
the number of lattice sites. There is no first-order
effect because (OlVlO& vanishes identically. However,
in second order we find contributions in which V acts
on l0& flipping a spin over. If the second application of
V in the matrix element (OlVgVlO& flips that spin back
to its original position, a nonzero contribution results.
It is shown pictorially in Fig. 15, and gives a contribution
-
Fig. (15) =[N/(-4+2h)](1/A. ) .
(4.50)
The crosses in the figure denote the action of the perturbation V and the vertical. line labels the flipped spin.
The energy of the intermediate state is
4+ (N —1)h —h.
(4.51a)
The four accounts for the fact that flipping one spin
breaks two bonds in Wo and each broken bond costs two
units of energy. The (N —2)h is just the energy from
the external magnetic field. The energy denominator
is then
Nh
—[4+(N- 2)h] =-4+2h,
(4.51b)
as stated in Eq. (4.50). There are no third-order contributions. In fourth order there are two distinct types
of contributions. The two flipped spins may or may not
be nearest neighbors. If they are not nearest neighbors
(Fig. 16a) four bonds are broken, otherwise only two
bonds are broken (Fig. 16b). Thus the energy denominators must be calculated separately. One must be
sure to sum over the different orderings of the vertices. These simple computations give
7C
5C
5
5i
(('
(a)
Rev. Mod. Phys. , Vol.
contributions to Eo(h). The horizontal. dotted li. ne means that
the two f1.ipped spins are nearest neighbors. The dashed
lines labeling intermediate
states have been omitted.
51, No. 4, October 1979
4N
(4.52a)
(I/A, )4
(4. 52b)
And finally there is the term
-(alVgV la&(alVg'V
la&
=
-
N
4
.(I/~)'
h
(4.53)
to the fourth-order coefficient c, of
Eq. (4.33b). The only effect of this last term is to
subtract off the N' term of Eq. (4.52) and render the
which contributes
vacuum energy extensive,
i.e. ,
Collecting these results,
1
proportional
to
¹
we have
4
6
(-4+ 2h)2 (-8+ 4h)
(-4+ 4h)
(4.54)
„8
and the magnetization
'
1
M = —~Eo
N h
series becomes
= 1 ——(I/A. )'
—
128
(1/A. )~ —~
.
.
(4.55)
Since this series does not terminate, we need a procedure to test whether it is "tending to vanish" at a
finite value of I/A. . According to earlier discussions,
we expect that in the critical region the magnetization
might vanish as a power of (A. —A, , ). We can test the
compatibility of this hypothesis with our series for M
by applying the ratio test (Stanley, 1971). The general
idea of the method is the following. Suppose we have a
series
f(x) =
g a„x",
(4.56a)
n
for a function f(x) which develops a power singularity in
the vicinity of a critical point x, .
f(x) - b(x, —x) "=bx, "(1 —x/x, ) ', x&x, . (4. 56b)
Using the binomial theorem,
power-behaved function is
f( (-b
the Taylor
series for this
—
rI(r+r( —)+
( )
r( + ("r(r(+(-() r
lt
(4.56c)
Xg
The power behavior shows up clearly in the ratio of
successive coefficients
(4.57)
R, =a, /a, , = (1/x, )[1+(y —1/I)] .
Therefore a plot of Rg vs 1// would produce a straight
line with an intercept x, and a slope y —1. In this way
an estimate of the critical point and the critical index
y can be made.
FIG. 16. Fourth-order
2N(N —3)
2h) ( 8 4h)
One ean also consider the
"linear
extrapolant"
E, = lRg —(l —1)R(, ,
which equals x, ' identically for a pure power.
(4.58)
In ad-
ditio~, the slopes
S, = (Rr -Rs-z)/[1/l —1(l —1)]r
(4.59)
John B. Kogut: Lattice gauge theory and spin systems
equal x, '(y —1) for a pure power.
Tabl. e II shows an application of the ratio test to the
magnetization series. Six terms in its series [three
beyond our illustrative calculation Eq. (4. 55)j are recorded there (Hamer and Kogut, 1979). We see that the
power-law hypothesis is exact& E, are all exactly
unity, giving &, = 1 as always. In addition,
S, = -1.125 (all I),
(4. 60)
tures. The magnetization
indicates that the lowtemperature phase spontaneously breaks the up- down
global symmetry of the system's Action.
The self-duality of the model allows us to view
the magnetization and order from another interesting
perspective (Fradkin and Susskind, 1978). Consider
the Hamiltonian
(1/A. )H+ h
so, if we use standard terminology
-
M
(1 —A.
P o, (n),
(4.66)
n
')
(4. 61)
we have
—P —1 = -1.125,
(4.62a)
that we used in the computation of the magnetization.
Applying the duality transformation equation (4.19),
we obtain
(1/A)H(a", A)+h
g v, (n)=H(g;
A.
')+h
n
which implies
(4. 62b)
So, we have the exact result
(4.63)
for all temperatures below T, (Pfeuty, 1970).
Of course, expansion methods do not usually yield
exact results. It just happens that in these examples
the exact results are simple functions of the expansion
parameters ~ and & ', so we could reconstruct precise answers. Applying the same calculational. methods
to the susceptibility and using the ratio test, one obtains an estimate of the critical index y (Hamer and
Kogut, 1979),
|..76*.03
|.
a =0.01+.02,
(4.65)
which compares favorably with the exact result of zero.
It should be remarked that the quantities E, and S, are
not very useful in these approximate calculations, since
subdominant power-behaved singularities make them
ra. ther irregular functions of I (Fisher, 1967).
It is hoped that the "dirty" calculations presented
here familiarize the reader with the "nuts and bolts" of
spin systems.
D. Kink condensation
and disorder
The magnetization M serves as a local order parameter for the Ising model. It is nonzero at 1.ow temperature, vanishes continuously as T passes through T„
and remains identically zero for all higher temperaTABLE II. Series analysis for the magn. etization of the Ising
model.
1
—0.125
—0 054 687 5
—0.034 179 69
~
-0.024 566 65
—0.019 039 15
Rev. Mod. Phys. , Vol.
R&
—0.125
0.437 5
0.625
0.718 75
0.775
1.0 (exact)
1.0
1.0
1.0
51, No. 4, October 1979
where we used the fact that
that
lJ,
, (m),
—1.125 (exact)
—1.125
-1.125
-1.125
g
, (n) = o(3n+1)cr, (n) to find
(4.68)
But first-order perturbation
that
&0I
p.
~, (~).
o. (~) =
o'. (~)l0)~=&01
n
theory in h now implies
g ",v, (m)I0). -&,
(4.69)
m &ff
where l0)„ is the vacuum of H(o; &) and l0)&-~ is the
vacuum of H(p. ;A. '). Since o, and o'3 are equivalent to
(4.69) tells us that the operator
g, and p,
„Eq.
(4.64)
to be compared with the exact result
75. Similarly the
specific heat exponent n is found to be
a&
~
m&n
(4.67)
P= 1/8.
y =
g
o, (n),
(4.70)
m&n
has a nonvanishing expectation value in the high-temperature phase of the original model. It is conventional
to call this operator a "disorder" parameter (Kadanoff
and Ceva, 1971). It has an interesting physical interpretation. Suppose it acts on a completely ordered
state having all spins "up, " v, (n) state) = lstate), for all
sites. Then it flips all%he spins to the left of siten. It
creates a "kink. " This observation exposes why the
expectation value of the disorder operator vanishes in
the low-temperature phase.
Since the energy of the kink state is localized in the
vicinity of site n, one can think of it as a particle
excitation. A zero-momentum kink state would be
l
l1 kink) =
p,
o, n)l0)z-& o.
(4.71)
m&n
It follows from the duality relation [Eq. (4. 67)] that the
computation of the kink's mass in powers of A. ' is
identical to that of the flipped spin states' mass in
powers of A. . Therefore the mass gap formula G(A. )
= 2l1 —A, l applies to the kink in the low-temperature
phase.
Since the vacuum expectation value of the disorder
parameter is nonzero in the high-temperature phase,
the ground state of that phase is a "kink condensate.
The short-range character of the spin-spin. correlation function can be understood intuitively in these
terms. Since the disorder operator flips spins over an
infinite region of space, a condensate of kinks random-
"
John B. Kogut: Lattice gauge theory and spin systems
677
Therefore, as we expand II(1+ a;o',. tanhK), only those
terms without a free spin variable will contribute. The
product II~;j& indicates that we place "bonds,
tanhAO';0&, on the links of a square lattice. Only if the
bonds form closed paths will the term contribute (only
then will all the o', 's occur squared). In addition, no
link can be covered more than once. Systematic rules
can be developed to allow very-high-order calculations
(Wortis, 1972). The first several terms are shown
pictorially in Fig. 17, and the expression for Z(E) is
izes the spin variables and leads to a finite correlation
length. It is also clear that kinks have a topological
significance. Applying the kink operator to the lowtemperature ground state produces a state which interpolates between the two degenerate ground states of the
low-temperature phase. Kinks alter the boundary conditions of the system by inspecting the spins at the
extremities of the spatial lattice one can determine
whether there are an even or odd number of kinks in
the system.
The concepts of topological disorder and kink condensation will become more significant as we continue.
"
—
Z(K)(cosh K)
'
~
= 1+N(tanhE)'+ 2N(tanhK)'
2
,
+ N(N
E. Self-duality of the isotmpic Ising model
It is interesting to use the partition function of the
Ising model to derive its self-duality. We shall do it
here for a square lattice and obtain the exact critical
temperature. The manipulations we shall. make also
constitute good preparation for analyses we shall do
later on lattice gauge theories.
Our strategy will consist of developing both high-
p
p
exp sc
where
E = J/kT,
"
p v;v&I,
Q
~ ~ ~
Q+j.
~&=
= (coshK)'«g
~pr
.
"
(4. 73)
tanhE = exp(-2K*)
it to Eq. (4.72)
Applying
Then comparing
Z(K*)
(coshK+~;&~sinhK)
(P«kP
&i j&
~ ~ ~
g
[
~
0
~
~
0
~
I
0
~
—
0
~
~
~
~
0
—
~
0
~
~
~
0
0 ——
~
0
0
~
I
~
~
~
Rev. Mod. Phys. , Vol.
(b)
51, No. 4, October 1979
.
~
.
(4.78)
Eqs. (4.77) and (4. 76) gives
Z(E)
2«(cosh2KP
sinh(2K) sinh(2K*) = 1,
(4.79)
(4.80)
and Eq. (4. 79) can be written
Z(E*)
.
s inh '(2 E *)
0
0
0
~
0
~
~
~
~
~
~
~
~
0
~
~
~
~
~
== 0
~
~
0I
0
0
0I
0
0
~
0
~
~
~
~
(a)
+
written
(4.75)
0
"
E
&ij&
v=o,
—
5)e
Note that Eq. (4.78) defines a mapping which takes large
into small A* and vice versa. There is more symmetry in these expressions than meets the eye. Manipulating hyperbolic functions shows that Eq. (4.78) can be
(1+ o;&& tanhK) .
The high-temperature
expansion now consists of using
Eq. (4.74) to develop a power series for Z(K)(coshK) '«
in the variable tanhE. To identify nonzero contributions, note the trivial identities
~
"«+ ~N(N-
(4.77)
(4.74)
~
=1+Ne ~+2Ne
The correspondence between the high- and low-T expansions should be clear. If the flipped spins of the
low-temperature expansion are surrounded by a square,
the graphical rules of the low-T expansion map onto
those of the high-T expansionl To make the relationship precise, define a coupling K* such that
j,
This is a very useful identity.
gives
Z(E) =
'
Z(K)e
the no'tation (ij) indicates a sum over
nearest-neighbor sites i and
and N is the number of
lattice sites. If K is very small, Z(K) can be expanded
in a power series. First observe that if 0 is a generic
Ising variable (&=+1), then
exp (Ko') = c oshK+ o s inhE
.
The factors of N record the number of ways the graph
can occur on the lattice.
Next consider the low-temperature expansion of Z(K).
Since all the spins are aligned at zero temperature,
this will be an expansion in the number of flipped spins.
Choose the T =0 ground-state spin configuration to be
all "up. When one spin is flipped, four bonds are
broken. If two spins are flipped, then six bonds are
broken if those spins are nearest neighbors and eight
bonds are broken otherwise. The first several terms
in the low-temperature
expansion are visualized in
Fig. 18, and the expression for Z(K) is
&i j&
N
~ ~
(4.76)
temperature and low-temperature expansions of the
partition function Z. These calculations will expose an
exact mapping between the high-T and low-T properties of the model.
Consider the partition function
z(tc) =
—5—)(tanhK)'+
—
—
(c)
0
0
—
Z(E)
s inh '(2, K)
(4.81)
~
FIQ. 17. Low-order contributions to the high-temperature expansion of &(~).
678
John B. Kogut: Lattice gauge theory and spin systems
0
~
FIG. 18. Low-temperature
expansion graphs for Z(~). A
cross indicates a flipped spin.
~
~
~
0
~
0
~
~
~
~
~
~
~
0
0 —-X- —~
0
—!
Xf
0
~
~
~
~
~
0
~
—
X X—
!
!
~
~
I
X
!
I
I
0
l
I
"
~
~
~
0
~
~
0
~
~
~
~
~
I
'I
~
~
0
(b)
So, these equations define a duality mapping which exposes a special symmetry between the model's highand low-temperature
properties. We can argue again
that if the model has a unique critical point it must be
at a temperature such that %=K*. Equation (4.80) then
yields the critical value
sinh'(2K, ) =1
(4.82a)
or
(4.82b)
This critical point was found long before the Ising
model was solved (Kramers and Wannier, 1941).
Note that Eq. (4.82a) is a special case of the critical
curve [Eq. (4. 15)] cited earlier. Using the same methods employed here and doing a bit more bookkeeping,
the reader can obtain that result.
The self-duality of the Ising model can also be
obtained without the aid of expansion methods. But the
expansions are of considerable interest for practical
calculations as well as rigorous analyses. One can
prove, for example, that the high- and low-temperature expansions of the Ising model in more that one
dimension have finite radii of convergence (Domb,
1972). Therefore the expansion alone teach us that at
sufficiently low but nonzero T the system magnetizes
in more than one dimension, while at sufficiently high
but finite T the system is disordered and the magnetization vanishes. Proofs that high-temperature
expansions have finite radii of convergence can be given
for a wide class of models. Analysis of this sort is
usually the first step one takes in trying to determine
the phase diagram of an unfamil. iar physical system.
We shall do such analyses for lattice gauge theories
later in this review.
F. Exact solution of the
Ising model in two dimensions
Finally, we want to solve the T-continuum formulation of the Ising model exactly. The point in doing this,
besides the illustration of interesting technical topics,
is to show that at the critical point the lattice theory
becomes a relativistic field theory: the spatial lattice
becomes irrelevant and the desired continuous spacetime symmetries are restored. We shall see that the
model becomes that of a free, massless, self-charge
conjugate fermion (Pfeuty, 1970). The explicit solution
illustrates the general strategy of using lattice formulations of quantum field theory: one first determines the
theory's phase diagram and critical points, and then by
approaching the critical points one constructs appropriately relativistic field theories.
Consider the Hamiltonian
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
H =—
Q o, (n) — Q o, (n)v, (n+1) .
(4.83)
A.
which is equivalent
to our previous expressions such as
Eq. (4. 14). There is a standard construction to expose
the fermion hiding here. Recall. the Jordan-Wigner
transformation (Jordan and Wigner, 1928) which accomplishes this. First define raising and lowering
operators
o'(n) =-,'[o, (n)+io, (n)],
(4.84}
a (n) =-,'[o, (n)-io, (n)].
Label the spatial lattice sites n = -W, -N+ 1, .. . ,
Then
fermion operators c(n) can be constructed from the
Pauli matrices
¹
n-1
]' exp[iso'(j
c(n) =
c~(n) = a'(n)
n-' j.
)cr
(j )]o (n)
exp[-iso'(j)o'
(j)].
The idea behind this construction is the following: The
Pauli matrices o+ and & anticommute on the same site.
Their squares are zero. In this sense they resemble
fermions. However, fermion operators anticommute
even when they have different spatial arguments. The
"strings" of operators in Eq. (4.85) are inserted to
insure this.
To check the claim that c(n) and c (n) are fermion
operators, it is best first to simplify the Pauli spin
algebra in Eq. (4. 85). Observe that
a
'[1 —a, (n)],
(n)o'(n) = —,
v'(n)&x (n) = —,
'[1+v, (n)]
(4. 86)
exp[i(m/2)o, ] =is, .
These identities allow us to write
n
l.
j="N
(4. 87)
n-1
c'(n) =~'(n) yJ,
[-,(j)].
&t is now easy to verify that the operators
c(n) and c (n)
satisfy fermion anticommutation relations,
(4.88)
(c(n), ct(m)] = 5„„(c(n),c(m }] = 0.
For example, using Eqs. (4.86) and (4.87), we see that
c(n)c (n)+ct(n)c(n) =o (n)o'(n)+O'(n)o (n)
(4.89)
In addition,
choose
m. & n
and compute
John B. Kogut: Lattice gauge theory and spin systems
c(m)c (n)+ct(n)c(m) =a (m)
n-l.
'
'
[-a, (j)]a'(n)
2n
2N+
n-j.
+ a'(n)
'~'
J[
[-o,(j)]a
(m)
.
(4. 90)
Using the fact that
a (m)a3(m)
=-a, (m)o' (m),
(4.91)
see that the right-hand side of Eq. (4. 90) vanishes
identically. The remaining parts of Eq. (4. 88) are
verified s imilarly.
Our next task is to wr ite H in terms of the f er m ion
operators. Using Eq. (4. 86) we have
(4.92)
a~(n) =2a'(n)a (n) —1 =2c (n)c(n) —1.
we
The coupling terms require a bit more work. Begin
1'
4m
2K+ 1'
2nN
'
2N+1
This choice implies a certain treatment of boundary
conditions (Schultz et. al. , 1964). Since we shall only be
interested in. the energy-momentum
relation and the
mass gap of the model, we need not be careful with
such points. We shal. l frequently discard constants as
we simplify H. The boundary conditions are not without interest, however, and the reader is referred to
the literature for a discussion (Schultz et. al. , 1964). Anyway it ls easy to ver1fy that the oper ators Q and Q
have fermion anticommutation
relations
(+«' o«)
5«', »
(a. .. a„] =(at, , at] =0.
Simply use the completeness
relations
with
a, (n)o;(n+1) =[a'(n)+ a (n)][a'(n+1)+o' (n+
1)].
(4.93)
Cons ider the pr oduct
c (n) c (n + 1) = o'(n) [-a, (n)] a
2%+1
(n + 1)
.
~
~
e&n(k- l)
(4.95)
so
c (n)c(n+ 1) =a'(n)o (n+1) .
(4. 96)
It is clear now that the coupling term can be written
in terms of fermion operators which are also only
coupled if they are nearest neighbors. The fact that II
can be expressed in terms of locQl products of the
fermion operators makes this change of variables useful. Note that the Ising character of the original degrees of freedom, a'(n) =1, was essential in leading
to this result. Other products of fermion operators
are computed as easily:
c(n)c (n+ 1) =-o (n)a'(n+1),
ct(n)ct(n+ 1) = a'(n)a'(n+ 1),
H
Next we must write H in terms of the
Since Eq. (4. 100) eau be inverted,
c(n)=
the computations
and
'""a, ,
ak.
(4. 104)
For example,
are straightforward.
g ct(n)ct(n+1)
P g P e'""e" ""a«a«,
=
n
n
~e
Qk
k
-k'
Qk
Qk'
(4. 105)
~
Collecting similar results for the other terms in Eq.
(4. 99) gives
(n+1)+c(n+1)]. (4. 98)
H=-2 Z
k
a»a»
—A.
(e
„+e-ik a«a»
a»a
k
becomes
g ct(n)c(n)
—
g [c~(n) —c(n)][c (n+ 1)+c(n+ 1)] .
+ e ~« ~«
=-2
A.
=-2
e
"cn,
where the complete set of wave vectors is
51, No. 4, October 1979
~ o«+-»)
P (1+A. cask)a»ta«- P (e '"a«tat» —e'"a«a
A.
k
(4. 106)
It will prove more convenient to write the sum over
modes just over those with k&0 and include the others
by simply writing them out. Then Eq. (4. 106) becomes
H
(4. 100)
«)
k
(4.99)
The solubility of the model is now clear, since II is
only a quadratic form. It is convenient to write II in
momentum space, so define operators
Rev. Mod. Phys. , Vol.
Ee
)
(
Qk
(4. 97)
Col. lecting all this we have
The Hamiltonian
Eq. (4. 102).
e -ik QkQ
c(n)c(n+1) =-a (n)a (n+1) .
o, (n)o;(n+1) =[c (n) —c(n)][c
that the fermion
relations for c(n) and c (n) imply
and verify by explicit calculation
anticommutation
(n) = -a'(n),
s
(4.94)
But
o'(n)a,
kel
g (1+A, cosk)(a»ta»+at»a „)
+2iA, g sink(a»tat»+a»a „) .
=-2
k&p
k&p
(4.107)
.
680
John B. Kogut: Lattice gauge theory and spin systems
Note that the vacuum of H is not the vacuum of the
operator ak because of the presence of (a»a „+a„a „)
in Eq. (4.107). We want to write H in the form
H=
Ak qk
X sink
(l+ X cos
(4.108)
gk+ constant,
k)
FIG. 19.
so'we can identify the single-pa, rticle excitations above
the vacuum (qk)0) =0). This ean be done by making a
canonical transformation from the operators a„, ak~ to
a new s et gk, gk~. Define
~k
=ukak+»ka-k
'gk
=ukak
~
O-k =uka-k
—'4vka k,
'g k
) = ~k ', k t
(4. 110)
(nk, nk) =(nk', nkj =0,
leads to a relation between
2
2 =
1
Qk+vk
k
uk~k
kl
k
Substituting
gives
H=
and qk
kn k
k
k, a
ak =uk&4 +zvkq
k
=uk'
into the Hamiltonian
g [-2(1+
,
+
A.
eosk)(uk
»
9-»)
k
+
1»
1-k)
k~k
(4.112)
and collecting terms
- vk) +
4A.
sinkukvkj
v'I + 2A.
Now we can return to
cosk+ A.' '
Eq. (4. 113) and evaluate the co-
efficient of the operator
yields
„).
Some algebra
so
(4. 121)
which is plotted in Fig. 20. Note that the minimum
value of Ak l.s obtazned at k =+ m where
in agreement with our earlier analysis. The critical
index v=1 as listed in Table I.
Finally, consider the theory in its critical region.
To discuss the encl gy —momentum relation we must
restore physical units. Since Ak is a minimum at
k =+a, we measure momentum from m. Let
=n+k'a,
(4.123a)
Eq. (4. 111) it is sensible to parametrize
coefficients with an angl. e &k,
.
Also, define
(4. 123b)
so that the energy has correct dimensions. Since we
want to consider finite values of k' as the lattice
(4.113)
In light of
of momentum.
X(k') =A»/2a
—v„')
~
~k
(q„@»+AD kq
{4.120)
so that k' has dimensions
that H have the form of Eq. (4. 108) implies
4(1+A. cosk)ukvk+2k, sink(uk —vk) =0.
(4.114)
6k, vk = sin
(4.119)
1+& cosk
k
Demanding
uk = cos
cosk + X2 '
(4.122)
—ivk
p [4i(l+& cosk)u„v»+2ih. sink(uk
(1» 9-k +
0'1 + 2A,
first invert Eq.
k&0
(0» k+ i
so that
X sink
and vk
(4. 111)
gk
sin 20»
(4.118)
A„=2 V'1+ 2~ cosk+ X2,
uk
.
To write H in terms of
(4.109)
4(1+A. eosk )2+ Amsin2k = v'1+ 2X cosk+ X2,
and we choose sign conventions
=uka k+&vkak,
which
This relation can be represented with a right triangle,
sketched in Fig. 19. The hypotenuse is
(4.109)
where k&0 everywhere. The functions uk and vk will
be determined by two criteria: (1). The qk, q„should
be fermion operators. (2). The Hamiltonian should be
diagonal. ized when written in terms of the qk, qk~. The
choice of the precise form of Eq. (4. 109) is made with
some hindsight, i.e. , the two criteria can be met if
uk and vk are real, even functions of k. The appearance
of i in Eq. {4.109) also leads to more elegant algebra
later. Anyway, it is easy to verify that the first
requirement
k 1» ', 1»
The parameters of the canonical transformation
diagonalizes the Ham iltonian.
the
spacing
a-0,
k
is forced to
m.
Then
Ak
simplifies
2A» = v'1+2K cos(m+k'a)+A. 2= (1 —A. )'+A. (k'a)'.
(4.124)
Ther'efore the energy-momentum
relation becomes
{4.115)
2 I+)
Then
—v,' = cos2 61k
so Eq. (4. 114) becomes
2ukv„= sin26»
uk
2(1+A. cosk) sin26»+2k, sink eos28» =0
(4.116)
(4. 117a)
or
+ sink
'
(1+%, cosk)
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
(4.117b)
FIG. 20. The energy-momentum
relation for the Ising model.
.
681
John B. Kogut: Lattice gauge theory and spin systems
E'- I/a as
A. is not in the critical region,
and we do not have a nontrivial continuum limit.
ever, if ~ =1,
So, if
a- 0
How-
(4.126)
and we have the relativistic spectrum of a massless
particle. We have seen that the particle is a fermion.
In fact, it is a self-charge conjugate, Majorana field
(Schultz et. a/. , 1964).
V. WEGNER'S ISING LATTICE GAUGE THEORY
A. Global symmetries, local symmetries, and the
energetics of spontaneoos symmetry breaking
Since one of our first steps in understanding a spin
or gauge system consists of mapping out its phase
diagram, it is useful to develop an intuitive understanding of how symmetries can be rea1. ized in these
systems. Our study of the two-dimensional Ising model
has shown us how a global symmetry can be spontaneously broken. At low T that system's ground state
is doubly degenerate. One state has positive magnetiza'tion, the other has negative magnetization.
Since these
two states do not mix through any finite order of perturbation theory, the spectrum of the theory must be
based on only one of the two alternatives. In this way
the space of states will not respect the up —down symmetry of the Action. Note that the symmetry which
spontaneously breaks down here acts on an infinite number of degrees of freedom occupying an infinite volume.
In fact, it appears that only symmetries of this kind
can break down spontaneously.
If a quantum system has
a finite number of degrees of freedom, then its ground
state is unique and symmetries of its Action are also
symmetries of that state.
To appreciate these points consider some simple
examples. The Hamiltonian of the nonrelativistic
hydrogen atom is invariant under rotati. 'ons, and its
ground state is also spherically symmetric and unique.
symNext consider a particle in a one-dimensional,
metric, double-well potential sketched in Fig. 21.
The system's Hamiltonian is invariant under the
-x. Classically the ground state is
operation.
the particle can sit in one of the
doubly degenerate
two minima. The symmetry is not respected by either
state. However, if the problem is treated quantummechanically, there is a finite probability that the
particle will tunnel from one minimum to the other.
x-
FIG.
21.
—
. Double-well
potential.
—
Z(block) = exp(-2PI
),
(5.1)
to the partition. function. But such blocks can occur
anywhere on the lattice, and for a given perimeter they
can occur in many shapes. Summing over shapes gives
I
)= y~exp(-2PI }+ ~ ~, (5.2)
Z(block of perimeter
where the factor p. counts blocks of various shapes
and p is a small number which can be estimated using
properties of random walks. As I. grows, blocks of
perimeter I. become less and less likely, and the magnetization is not destroyed at sufficiently low temperatures. However, if the temperature is large enough,
the entropy factor p, in Eq. (5.2) grows faster than the
temperature-dependent
exponential falls. Then large
blocks are not suppressed, fluctuations are important,
and the magnetization vanishes.
The energetics of this example should be contrasted
with the one-dimensional Ising model. That model is
triviall. y soluble, and one knows that it is magnetized
only at T =0 (Stanley, 1971). In fact, one can prove
FIG. 22.
A block of flipped spins.
spins pointing up, (down).
.
Rev. Mod. Phys. , Vol. 5't, No. 4, October
This effect lifts the classical degeneracy and produces
a symmetric ground state. The ground state becomes
the symmetric combination of the alternative that the
particle reside either in the left or right well. The
antisymmetric combination has an energy which is
higher by an amount proportional to the tunneling
amplitude. The symmetric state has the lower energy
because its wave function is smoother
it has no
nodes.
It is interesting to think of the two-dimensional Ising
model as consisting of a double-well potential at each
site. Nearest-neighbor sites are then coupled together
in the usual way. Now the symmetry can be spontaneously broken. Let us ignore the quantitative analysis
we have done earlier which exposed this fact, and try
to understand it in crude terms. Imagine the system
with all spins up. One spin may flip by tunneling
through a finite potential barrier (four bonds are
broken). Such fluctuations will happen at low temperature, and they are responsible for a smooth decrease
of the magnetization with increasing T. Now consider
other fluctuations which tend to disorder the system
more. I.et a whole region of spins flip and suppose the
region has a perimeter I-, as shown in Fig. 22. The
only broken bonds are on the suyface of the block, so
this configuration contributes a term
1979
~
The pluses (minuses) denote
John B. Kogut: Lattice gauge theory and spin systems
that spin systems with nearest-neighbor couplings
and discrete global symmetries can only experience
spontaneous breaking at nonzero temperature in more
than one dimension (Stanley, 1971). We can understand
this by considering a block of length L of flipped spins
as shown in Fig. 23. Since only two bonds are broken,
such a block contributes a finite term to Z for any
length L. If L- ~, so that we have a kink configuration, an infinite number of spins are flipped with onl. y a
finite cost in Action. This means that for T ~0 the
ground-state expectation value of the magnetization in
zero and the global symmetry is restored. Clearly
the energetics of this model resembles that of a
single doubl. e-well potential. This point ean be made
pr ecis e by c ons ider ing the 7 - continuum Hamiltonian
formulation of the one-dimensional Ising model. Then
the quantum Hamiltonian acts on only one spin variable. Foll. owing the discussion of Sec. IV.A we easily
obtain the Hamiltonian
(5.3)
and s olve for its spectrum
(5.4)
The up
down symmetry of the model is realized
through the spin-fl. ip operator o„
(5. 5)
This is clearly a symmetry of the spectrum Eq. (5.4).
The ground state is unique and symmetric.
These examples illustrate the importance of dimensionality of the physical system to its phase diagram. Another essential ingredient is the dimensionality of the local degrees of freedom occurring in the
Action. Later in this article we shall study theories
with a continuous global symmetry
S
=-J
s
n
~
sn+p. ,
(5.6)
n, P
where s(n) is a. two-dimensional unit vector in the
"planar" (or 0, ) model and a three-dimensional unit
vector in the 0, Heisenberg model. There are rigorous
theorems (Mermin and Wagner, 1966) which state that
the global continuous symmetries of such models cannot
break down spontaneously in two or fewer dimensions.
We shall understand this result later after some detailed analysis. Roughly speaking, these models do not
magnetize in two dimensions while the Ising model does,
due to the presence of smoothly varying spin configurations. They tend to disorder the system and their
multiplicity is so great that they succeed in two dimensions at any nonzero temperature.
Qne says that
"critical dimension" of the O„models is two because only above this number of dimensions will they
have two phases. The critical dimension of the Ising
model is one.
the
B. Constructing an Ising madel with a Ioeal symmetry
As a first step toward lattice gauge theory formulations of the strong interactions, we shall consider Ising
lattice gauge theory. We want to take the degrees of
freedom of the Ising model and couple them together
in such a way that the global. symmetry of the ol.d
model is elevated to a l'ocal symmetry. When this
construction is generalized to systems with continuous
symmetries, we shall recognize the models as lattice
versions of theories with gauge symmetries.
F. Wegner invented Ising lattice gauge theory in 1971
(Wegner, 1971). His motivation was to obtain models
which could not magnetize but, would have nontriviaJ.
phase diagrams. His inspiration was the I'l. anar model
in two dimensions.
It was known that such a system
could not magnetize, but it was suspected that the theory had a phase transition because high-temperature
expansions of the theory's susceptibility indicated a
singularity at a reasonabl. e temperature (Stanley and
Kaplan, 1966). The idea of a "phase transition" without a l. ocal order parameter is quite novel, since it
challenges us to find a symmetry of the system which
can distinguish the two phases. We shall see that
lattice gauge theories pose similar conceptual problems, and Wegner's work sheds much l.ight here. He
realized that endowing the lattice system with a local
invariance group forbade the occurrence of a magnetization, since a local symmetry cannot break down
We shall prove this fact (Elitzur, 1975)
spontaneously.
in the next section. Wegner also showed that his models
had phase transitions, and he suggested how the various
Although his
phases could be labeled and distinguished.
work is not generally cited, his 1971 paper ranks among
the most significant in the fiel. d.
Consider a cubic lattice in d-dimensional Euclidean
space-time. Label. links of the lattice by a site n and
a unit lattice vector p. . The same l. ink can be labeled
as (n, p, ) or (n+ p, , -p, ). Place Ising spins (v, =+1) on
l. inks.
Define a l. ocal gauge transformation at the site
n as the operation G(n) of flipping ail the spins on links
connected to that site. An example is shown in Fig. 24.
The Action has a huge invariance group because G(pg)
can be applied anywhere. A nontrivial. Action having
this symmetry consists of the product of spins around
of the lattice,
primitive squares, or "plaquettes,
"
S=-j
o n p
o3n+p,
v
x o, (n + p. + v, —p. )o, (n+
v,
—v)
.
(5.7)
The arguments of the spin variabl. es label the relevant
l. ink.
We shall sometimes use the generic notation
~3~3~3~3
(5.6)
for easy presentation.
FIG. 23.
A blocl~
of flipped spins in the one-dimensional
madel.
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
Ising
We must first check that S is invariant under arbitraIf the operation G is
ry local gauge transformations.
John B. Kogut: Lattice gauge theory and spin systems
G(n}
0p (2}
FIG. 24.
theory
I.ocal symmetry
three dimensions.
A
in
local as possible.
If we want to discuss the energetics of Ising lattice
gauge theory, we must use a language which respects
the l.ocal gauge invariance of the Action. Recall that
when dea. ling with the ordinary Ising model we could
speak of broken. or unbroken bonds concepts which
are invariant to the global symmetry of that model's
Action. However, the concept of a broken bond is not
locally gauge invariant. For example, consider the
plaquette of spins shown in Fig. 25. Applying a gauge
transformation at n changes the character of the bonds
o'»(1)o', (2) and a'»(3)os(4). So, it is only the relative
orientation of four spins around a plaquette that is
gauge invariant. There are two possibilities:
—
o, (i) =1
&o. (~, v))» =
(w, v)
expI yea, z v, v
expIppa,
v, v, u, +A;pu,
I
I
j..
(5.9b)
g o, =h g o,'-h P
where o,' is a transformed
6cr,
(5.11)
,
spin and
5o, (l„) o,'(l„) —o, (l„) = -2a, (I„),
5o, (I) = 0 if, / & {/ .
case the spins are "frus1976), while in the first case they
One says that in the second
=-
(5.12)
j
03- &3'
Making this change of variables
impossibility
+ape
(5.10)
Jz
—the
-'-'-
%'e shalt. use the local gauge invariance of the Action
in the absence of @ to show that the right-hand side of
Eq. (5.10) vanishes smoothly as h- 0. Consider a local
gauge transformation at the site n. Denote the set of
links emanating from site n by {l
The four-spin term
in the Action is invariant to this gauge transformation,
but the external field term becomes
or
C. Elitzur's theorem
r
'-—
SPIn COnfjg
(5.9a)
trated" (Anderson,
are not.
of spine.
this result is certainly plausible. The proof is not
diff icul. t.
To test whether a system sustains a spontaneous magnetization, , one places it in an external field h, which
couples through a term hQ„„o,(n, p), computes
&o'»(m, v)), and then takes the limit h- 0. If the expectation value of &3 is nonzero in the limit, the system is
magnetized. For lattice gauge theory, a nonzero expectation value would imply the spontaneous breaking of
not just the global up =down symmetry, but the local
symmetry as well. Consider the expectation value
applied at the site n, then both spins o, (n, p. ) and
o»(n+ v, -v) = &, (n, v) change sign. Therefore 8 is unchanged. A little thought shows that the essential in. gredient in guaranteeing the gauge invariance of 8 is
that it be constructed from the products of o3 taken
around closed paths of links. In constructing S one
chooses primitive squa, res of four links to keep it as
"
A plaquette
(!)
0
operation in Ising lattice gauge
FIG. 25.
of spontaneously
in the
expres-
sion Eq. (5.10) gives
breaking a local symmetry
%e want to
&o', (n,
lattice gauge theories can
have interesting phase diagrams and, if so, how one can
label their phases. The first point to make is that
transitions with local order parameters cannot occur.
This is a consequence of Elitzur's theorem, which
states that the spontaneous magnetization of this model
vanishes identically at all T as a consequence of the
local gauge symmetry (Elitzur, 1975). In light of our
intuitive discussion of spontaneous symmetry breaking,
l&o»(~
know if Ising
v))»-&-o»(~~ v))»l =
-o.(~, v)
exp
-i Q &o.
{s }
v))„=- Q &,'(n,
X eXP
03
n,
O303'O3O3'+
v
Now we can produce
—1
v)
exp -h,
h
5 03
a3'+ lZ
a
50'3
Z
(5.13)
a bound,
-le""-Ill&o.(~, v))»l,
(5.14)
n
where d is the dimension of space-time. But the righthand side of Eq. (5.14) approaches zero as h-0, so
&o, (&,
v))»-, = &-o,(~, v))»-„
which implies
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
(5.15)
&o'3(n,
v)) = 0
(5.16)
as claimed.
What was the essential ingredient in this proofs The
point is that even in the presence of h, spin configura-
684
John B. Kogut: Lattice gauge theory and spin systems
tions eonneeted by the operation G(n) differ in Action
only by a finite amount. Therefore only a finite energy
barrier is set up between two spin configurations, one
of which has v, (n, v) positive and the other v, (n, v)
negative. Thus the energetics resemble a quantum
mechanics problem involving only a, few local degrees
of freedom, and we are familiar with the fact that in
this setting symmetries do not break down sponta-
neously.
Elitzur's theorem applies to a wide class of theories.
Theories with a continuous local symmetry group also
cannot have a spontaneous magnetization.
The proof
given here easily generalizes to these cases, which
will be studied in later sections of this review. The
theorem can also be applied to the expectation vat. ues
of more complicated operators. The expectation value
of any operator which is not invariant under all local
must vanish identically. Acgauge transformations
cordingly we shall restrict our attention to gaugeinvariant operators from here on.
x3 (l)) ~ exp( —p).
(5.20b)
~
C
x
~
exp(pv, v, v3v, ) =coshp+ v, v, v~v, sinhp
= (1+ v, v, v, v, tanhP)
v, (l) =
P
splfI
C
config
—how
tes
(1+ v, v, v, v, tanhp)
to
label the phases of a theory having a local symmetry
group. Wegner suggested that we consider the spatial
dependence of correlation functions. His inspiration
was the two-dimensional planar model, which undergoes a phase transition without the appearance of a
The two-phase character
spontaneous magnetization.
of' the model. shows up in the system's spin-spin correlation function. At low temperature spin-wave
analyses show that the correlation function falls off
as a power,
(5.1'7)
(s(0) s(n))- ~n( "
~
while high-temperature
expansions show that the
havior is exponential for T sufficiently la, rge,
(s(0) s(n)) —exp{- ~n j/g(T)) .
be(5.18)
These estimates mean that the correlation length is
finite at high temperatures and is infinite at low temperatures. A phase transition must occur in. the intermediate temperature region. More powerf'ul analysis
is necessary to probe the details of that region of the
phase diagram.
Wegner invented and studied a gauge-invariant correlation function for Ising gauge theories. Since one
must consider the product of spin va, riabl. es around a
closed path of links, it is natural to consider
v. (I),
(5.19
e
where the arguments
coshP,
(1+v,v~v, v, tanhP)
piquet
correlation functions
Now we can return to our original. question
"'
~I~
So, at high T the correlation function falls very quickly
as the l, oop is taken larger and larger, whil. e at l. ow T
it falls off at a qualitatively slower rate. This result
proves that the system has distinct high- and lowtemperature phases.
It is easy and instructive to obtain these results. The
methods employed are simple generalizations of the
high- and low-temperature expansions of the Ising
model reviewed in Sec. IV. E. First consider high T.
%'e use the identity
I, E
D. Gauge-invariant
~I
I~
~II
lE=
c
.
(5.22)
If P is small compared to unity, it is sensible to expand
Eq. (5.22) in powers of tanhP as in See. IV.E. The fact
that
Q
v,
=o,
g v', g
=
g3= +]
1=2,
(5.23)
@3= kg
means that enough powers of v, v, v, v, tanhP must be collected from the expansion of the Action so that no
isolated factors of &3 belonging to IIco3 are left unmatched. A little thought indicates that the first nonvanishing contribution in the numerator of Eq. (5.22)
occurs when there is a plaquette of operators @3030303
for each square of the minimal surface bounded by C,
v
= (tanhP)
p+
(5.24)
~ x ~
c
where N, is the number of squares on that surface.
Eq. (5.24) confirms Eq. (5.20a) because it ean be
written
(c
=exp lntanh
A.
+
~
~,
But
(5.25)
If we calculated to higher orders, the "area law"
found here woul. d remain, but the coefficient lntanhP
would become a more complicated expression,
= exp
—
A
(5.26)
rf=-
of the spin variables denote links
and C is a closed contour. The expectation value of
such an operator will depend on the characteristics of
the contour C. In particular, the loop has a certain
perimeter
I', a certain
minimal
enclosed surface area
high T
A. Wegner showed (Wegner, 1971) that at
03 l
-exP -A.
l&C
while at low T
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
(5.20a)
where the leading term in the expansion of f(P) is
lntanhP. The point is that the "area law" holds for
finite but large temperatures.
The methods used to
prove that high-temperature
expansions of simpler
Ising systems have finite radii of convergence apply
also to Ising gauge theory.
Now consider low temperatures
and suppose that
d & 2. To develop a low-temperature expansion for
the correlation function it is convenient to organize
the sum over spin configurations in a particular way.
John B. Kogut: Lattice gauge theory and spin systems
of spins fo'3(n, v)j may be transformed
to another configuration (o,'(n, v)j by applying gauge
transformations at various sites. We have seen that
the physics of the model consists only of its gaugeinvariant content, so the configurations (o', (n, v)} and
" A gauge(&3(n, v)) are said to be "gauge equivalent.
invariant configuration of the system consists of a
spin configuration (o', (n, v)) and all of its gauge-equivalent copies. It is often convenient, however, to label
a gauge-invariant conf iguration by one of its r epr esentative configurations of definite spins. When
computing gauge-invariant
expectation values we can
do this with no loss of genera, lity or rigor. In such a
computation the sum over all spin configurations is
'replaced by a sum over representative spin configura-.
tions labeling different gauge-invariant configurations.
When this is done, a common multiplicative counting
factor is removed from the numerator and denominator
of an expression such as Eq. (5.22). In some of our
illustrative calculations which follow, we shall often
use the language of "flipped spins" and "representative
spin configurations" rather than the expl. icitly gaugeinvariant concepts of "frustration" and "gauge-invariant configurations. " In particular, it is helpful
when discussing the low-temperature
expansion to
choose the representative spin configuration at T =0
to have "all spins up" so that the expansion proceeds
in the "number of flipped spins. " So consider the expectation value
A configuration
(5.27)
The first term in the expansion is unity because
II,o'3=1 when all spins are up. Now let there be one
spin flipped. Then 2(d —1) plaquettes are frustrated,
so the Action of this configuration measured rel. ative
to the Action of the "all spins up" configuration is
4(d —1)p. Note that only if the flipped spin occurs on
the contour C will the numerator be different from
unity. In those cases it is -1. If N is the number of
links of the lattice and L is the number of links on
the contour, we have
,
o.
(
C
=
1+%—2L exp-4d-1
+~ ~
~
(5.28)
).
let's
Now
estimate the contribution to the numerator
due to n spin flips. Treating them as completely in-
(1+N exp[4(d —1)P]+
we have
dependent,
1
—
—
S
(
(N
~ ~
2L, )" exp[-4z(d
—1)P] .
The n-spin-flip contribution
itself is
(5.29b)
over n,
o, = 1+
C
exp
-4 d -1
'(N —2L)' exp[-8(d —1P)] + ~
+ —,
~ ~
)/Z,
(5.30a)
Rev. Mod. Phys. , Vol.
.
(5.30b)
So, both numerator and denominator exponentiate,
dependence on. N cancels a, s expected, and
the
(5.31)
and we have verified the "perimeter law, " Eq. (5.20b).
If m0re graphs were summed we would find that
o
L,
=exp -h
(5.32)
C
where the leading term in a low-temperature series
for h(P) is 2 exp[-4(d —1)P]. To calculate h(P) systematically one should develop a connected graph formalism (Wortis, 1972). Then one could take account of the
excluded volume effects which we ignored when discussing the n spin-flipped configurations in Eq. (5.29).
All this can be done systematical. ly, a.nd it does not
affect the leading term in the series for h(P). Since
this low-temperature expansion has a finite radius of
convergence, we have learned that the system has a
low-temperature phase which is distinct from its
high-temperatur e phas e.
Our argument for the "perimeter law" does not apply
In that case the low-temperature
in two dimensions.
expansion has a vanishing radius of convergence. There
are several ways to see this, and we shall discuss two
of them which a.re instructive. First observe that the
energetics of the two-dimensional model is very
special. If a single spin is flipped, then two plaquettes
are frustrated. However, if a line of spins are
flipped, then again only two plaquettes are frustrated—
those at the ends of the line, as shown in Fig. 26. So,
there is a special degeneracy problem in two dimensions which "disorders" the system at any nonzero
temperature and leads to an "a,rea law" as we shall
now see. This degeneracy problem does not occur in
higher dimensions because then plaquettes which come
out of the plane of Fig. (26) are frustrated by the line
of flipped spins. Let us calculate the gauge-inva, riant
correlation function by taking into account only configurations of flipped spins in which one of the ends of
the lines extends to infinity. Such configurations a, re
very efficient at disordering the system, and we shall
see by comparison with an exact calculation that they
are a good guide to the qualitative character of the
+
~
N-2L
~ ~ ~
to the partition function
1
(
Z=—1+N exp[-4(d —1)P]+ ~N2 exp[-8(d —1)P]+
(5.29a)
—
~N" exp[-4n(d —1)P) .
Summing
where
61, No. 4, October 1979
+
+
+
~
+
~
+
+
+
~
f
~
+
~
+
+
+
+
~
~
+
+
+
+
~
~
o
+
+
+
+
~
~
~
+
+
f
+
+
+
o
+
+
+
FIG. 26, A line of flipped spins and the accompanying
in two dimensions.
ed plaquettes
f
+
e
+
frustrat-
John B. Kogut: Lattice gauge theory and spin systems
This condition does not labe1. the representative spin
configuration of a gauge-invariant set uniquely. In fact,
within the temporal gauge the theory is invariant to. a
huge set of local symmetry operations. These consist
of al. l w-independent gauge transformations.
Consider
the application of such a transformation on a particular
column of sites of definite spatial position. The column
runs from v =-~ to v+ as sketched in Fig. 28. The
point is that since each spin on a temporal link is either
not affected by the transformation or is fl. ipped twice, the
condition Eg. (5.36) is maintained.
To obtain the r-continuum Hamiltonian, begin with the
Action written with anisotropic couplings. Let P, be
the coupling for plaquettes containing some temporal.
links and let P be the coupling for the remaining cases,
Eg. (5.47) is now purely spatial, of course.
The vector n labels sites and i labels the unit vectors
of the lattice.
Our first task is to understand the local symmetries
of the theory using the operator formulation. It is
easy to construct a v-independent operator G(n) which
flips the spins o'3(n, i) on all the links emanating from n,
underlying
G(n) =
'
'
o, (n, i).
For example on a three-dimensional spatial lattice +i
runs over six values, as shown in Fig. 29. Since 0,
is a spin-fl. ip operator for O„we have
(5.42)
G
'(n) &, (m, i )G (n) = o'„(m, i ),
G
"(n)o, (n, i)G(n)
=-o, (n, i),
G '(n)&, (m, i)G(n) =&, (m,
where P, and P, denote the two different types of
plaquettes. The first term simplifies because of the
choice of gauge
-p,
g o, (n, x)o, (n+ 7, x) .
(5.43)
{s~)
Aside from an irrel, evant additive constant,
can be written
'P,
—,
g [o,(n+0, x) —o,
(yg,
{s &)
this term
Q
.
(5.44)
[o,(n+r, x) —cr, (n, x)]' —p
Q
{P~
G '(n)ffG(n) =~,
(5.50)
so the Hamiltonian has the desired local invariance.
Elitzur's theorem implies that the space of states QP)]
is invariant to these local operations,
-~, P-xexp(-2P, ),
(5.46)
Hamiltonian
(5.47)
where the o', and e, are now operators.
This statement subsumes the fact that the system
cannot magnetize. In particular, consider the matrix
element
(glo, (n, i)lg) =(JIG(n)G '(n)o, (n, i)G(n)G '(n)lp)
=-(4 lc, (n, i)lq),
)
This expression should be compared to Eg. (4.2) of the
two-dimensional. Ising model. The construction of the
transfer matrix, a v-continuum limit, and a quantum
Hamiltonian parallels that discussion. The same limiting procedure
applies and leads to the quantum
i),
where the last formula applies only if the link (m, i)
does not coincide with any of those originating from
site g. It follows that,
o, o, cr, o, .
(5.45)
P,
(5.49)
(5.51)
x)]',
so
'p,
S = —,
(5.48)
(5.52)
which implies that
(5.53)
Now we can study the three-dimensional
Ising gauge
theory in detail. It is easy to see that it is dual to the
ordinary three-dimensional
Ising model. . To do this
we follow the same strategy used to see that the twodimensional. gauge theory was trivial. ; write the theory
in terms of a minimal number of degrees of freedom
by eliminating all the residual gauge freedom. If we
always work within the gauge-invariant subspace Eq.
(5.51), we have an operator identity
The lattice
G(n) =
o, (n, i) =1.
(5.54)
So
&, (n, $)&, (n,
-P)&, (n, x)o, (n,
-x) =1,
(5.55a)
which allows us to solve for o;(n, y),
FIG. 28. A v'-independent
gauge transformation.
FIG. 29. The links affected by
the gauge transformation G4'n).
Rev. Mod. Phys. , Vol.
51, No. 4, October |979
John B. Kogut: Lattice gauge theory and spin systems
688
v, (n, y) = c, (n, x)cr, (n, -x)o', (n,
-P) .
(5.55b)
But cr, (n, -$) can be treated similarly by considering a
gauge transformation at the site g, —y. Iterating this
procedure, one can write o, on any j) link just in terms
of those on x links,
v, (n,
)) = v, (n, x)c, (n, -x)&, (n —g, x)&, (n —X, -x)
x o, (n —2g, x)o, (n —2y, -x). . . .
(5.56)
This procedure means that we are treating 0, on links
which point in the y direction as dependent variabl. es.
To be consistent, the spin variables o', (n, +P) must also
be eliminated from the collection of independent degrees of freedom of the theory. In particular, H will
have no operators which do not commute with o, (n, g).
Therefore, choose it to be a constant,
v, (n, y)
=1.
they a, re independent.
The dual. ity mapping can now be defined. Associate
a site n* of a "dual. lattice" with each plaquette of the
origina. l lattice. Define a "dua, l spin-flip" operator on
this site,
p.
,(n+) = cr, o, c,cr, ,
(5.58)
where the four o, 's are those of the plaquette associated with n*, as shown in Fig. 30. "Dual spin" variables are defined by
p.
,(n*-x) =o, (n, y),
(5.64)
can be recognized as a consequence of Eq. (5. 56).
Therefore the Hamiltonian is
&s
—
=A.
n*
&3
n*+~
—g
p. z n+
jL
A.
, (n*) = n'~pJ' c, (n-n'g, x).
(5.59)
As in the case of our discussion of the two-dimensional
Ising model, the dual variables satisfy the Pauli algebra, and the Hamiltonian can be written simply in terms
of them. It is clear that
(5.60)
p,
(5.65)
which we recognize as the quantum Hamiltonian formulation of the three-dimensional. Ising model at a temperature A, *=A, ',
H(3-D Ising gauge;
A.
) = Afl(3-D Ising model;
A.
') .
Therefore we have a mapping between the high- (low-)
temperature properties of the gauge system and the
low- (high-) temperature properties of the spin system.
A great deal is known about the three-dimensional
Ising model. It is a two-phase system with a spontaneous magnetization labeling the ordered phase. It
undergoes a continuous phase transition at T, with
power-behaved singularities. We can now use the fact
that this model has a, local order parameter to l.earn
how to label the phases of the gauge theory. Since
phase of
{Ol p. , (n*) IO) is nonzero in the low-temperature
the Ising model, consider the Hamiltonian in an external field,
H(3-D Ising model;
A,
g
*) +h
(5.61)
terms of 0, and
a'3, have one link in common where a, a', =-o, o, holds.
Finally,
when written
in
p, (n*)p, , (m*) =g, (m*)p, (n*) (n*& m*),
tors, note that
(5.63)
follows trivially from Eq. (5.59). In addition
.
(5.6Va)
0~ and &3,
(1/A. )H(3-D
Ising gauge;
A.
)+h
g,
v, (m
-ny, x) .
(5.6Vb)
So, first-order perturbation
theory implies the
equal. ity
mode]
(5.62)
since the identity 0, &, =-0,0, must be applied an even
number of times to interchange the two dual operators.
To write the Hamiltonian in terms of the dual opera-
p, 3(n*)
This operator ean be written in terms of the variables
Also,
since these operators,
n+
(5.57)
o, (n, R) and v3(n, x) are the only variables in H and
Now
,
p. (n*)Lj,
.... ... .
=«I ", ', , { — y, ")IO)I ,
1
n-p
(5.68)
Since gs(m*) is an order parameter for the Ising model,
and since the duality mapping interchanges high and low
temperatures, we can identify a nonlocal disoxdex
parameter for the gauge theory,
', &, (m-ng, x)IO) =0
{Ol
(&
large)
(&
small).
p
{Ol, jo', (m-ny,
ft
&0
(5.69)
'Zhe similarity of these statements to our discussion of
the two-dimensional Ising model should be noted. Vfe
can think of
oomn$, x) as a kink operator. Equation (5.69) states that the low-temperature ground state
of the gauge theory is free of kinks, while for all T
above a critical point there is a kink condensate. Note
II„(
FIG. 30. The dual lattice sites
in two spatial dimensions.
Rev. Mod. Phys. , Vol. 5't, No. 4, October
&)IO)
Q
1979
689
John B. Kogut: Lattice gauge theory and spin systems
also that the gauge-invaxiant characterization of the
operator is stated in terms of frustation. If the operator is applied to the T =0 state of the gauge theory,
which is free of frustrated plaquettes, then a frustration is made at the plaquette m*. The phase transition
in the model can therefore be viewed as "frustration
condensation.
Since frustration and kinks have a
topological significance, we have exposed a symmetry
criterion to label the system's phases. Of course, the
kink operator is a no@local object, so it can have a
vacuum expectation value and not violate Elitzur's
theorem. The kink condensation idea also exposes the
8=- J
(6.3)
cos 4„0
(6.4)
rg
This Action has a global, continuous symmetry, i.e. ,
rotate all the spins through a common angle n. S remains unchanged since it is constructed from inner
products.
Now construct a theory which elevates this continuous
symmetry to a local continuous symmetry. We shall do
this by generalizing Wegner's Ising gauge theories in a
direct fashion. Experience with electrodynamics is also
helpful here. Place planar spins on each link of a lattice. Suppose that there are local symmetry operators
at every site n, which rotate all the spins on the links
originating from site n by a common angle y(n). This
should be an exact symmetry of the theory's Action. To
state this precisely let 8„(n) be an angular variable on
the link (n, p, ). Since the jink (n, p, ) can also be labeled
(n+ p, —p. ), we need a definition of the variable
(n+ p). We choose
8,
8
„(n+ p) = —8, (n),
(6.5)
for reasons which will be explained shortly. With each
of the lattice, associate the following "discrete curl":
plaquette
.(n)
8.
=
~.8„(n) ~.8. (n)
=8.(n+ p. ) —8, (n) —8, (n+ v) +8, (n)
(6.6)
=8, (n)+8, (n+ p)+8, (n+ p~ v)+8 „(n+v),
where we have used Eq. (6. 5) to express the curl as the
sum of angular variables around the dA ected plaquette,
as shown in Fig. 31. The interesting characteristic of
the curl is that it is invariant under local gauge rotations. Consider a gauge trans formation at the s ite n,
VI. ABELIAN LATTICE GAUGE THEORY
A. General formulation
F. Wegner's Ising lattice gauge theories were generalized to continuous, Abelian gauge groups by K. G.
Wilson (Wilson, 1974). Lattice gauge theories were
independently invented by A. M. Polyakov (Polyakov,
1975). The quantum Hamiltonian approach was developed
by the author and L. Susskind (Kogutand Susskind, 1975).
I et us begin the discussion of Abelian models by recalling the character of spin systems with a global
Abelian symmetry group. Place a planar spin on each
lattice site n,
cos8(n)
—8(n),
n, v
physical mechanism which makes the gauge-invariant
correlation function satisfy the area 1aw at temperatures above T, . If we consider a purely spatial contour C and compute (O~IIco, (0) in the approximation that
treats the high-temperature ground state as a kink
condensate, then the area law is obtained in the same
way that we found the area law for the two-dimensional
Ising gauge theory in Eq. (5.34).
Before moving on to our next topic, let us review
some facts about the four-dimensional Ising gauge
theory. Using methods we have already discussed,
one can show that the theory is self-dual (Wegner,
1971). The self-dual point is X =1 in the r-continuum
Hamiltonian language. Expansions for the theory's
free energy indicate that it is a two-phase system with
a first-order phase transition at A. = 1 (Bajian et al. ,
1975). This fact has been confirmed by computer simulations (Creutz, et. a/. , 1979). It has been suggested
that the theory is soluble (Polyakov, 1977) and can be
diagonalized in terms of "fermionic string" variables.
Research in this direction is being actively pursued by
several groups.
'
p, )
so that Eq. (6.2) becomes
"
s
8(n)=8(n+
&
8. (~) —8. (n)
q(n)
.
(6. 7)
Equation (6.5) implies that under this gauge transformation
8, (&
p)+-8 „(&+p)+X (&)
(6.8)
Therefore it is clear that 8, „(n) is invariant to this
operation. We can make this invariance principle appear more familiar by considering simultaneous gauge
transformations one at site n involving the angle y(n)
and one at the site n+ p, involving the angle y(n+ p, ).
Then 8 (n) transforms as
~
(6. 1)
0
sin8(n)
Couple nearest neighbors
8
together in the traditional
fashion,
s
=-ZQ
n, v
Ã
s tl+
p
(6.2)
8 „(n+ v) ~~
(n+p+
v)
~
n+/4 + 1/
ii
8„(n +p. )
cos[8(n) —8(n+ p)].
This can be written more elegantly if we introduce a
finite difference operator in the direction p. ,
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
.
FIG. 31.
A
directed plaquette and its angular variables.
690
John B. Kogut: Lattice gauge theory and spin systems
8, (~)-8„(n) —x(n)+x(n+ p)
-8 (n)+ 4 x(n),
(6. 15)
(6.9a)
while the curl is invariant,
8., (n) —8„„(n).
(6.9b)
These equations are recognized as a discrete form of
the local gauge invariance of electrodynamics,
A.
-A„+8
X,
E, -F „.
(6.9c)
So, to make an Action which is locally gauge invariant,
we use the variables 8 „(n) and define
1 —cos8,
S =&
n, gv
„n
(6. 10)
This Action has two important symmetries. It is locally gauge invariant because it is constructed from
the lattice field 8„,(n). It, is also a periodic function of
8„„(n), in the same way that the simpler spin model
Eq. (6.4) is a periodic function of & 8(n). We are in=
terested in this type of Action for reasons which will be
discussed when non-Abelian gauge theories are introduced.
Before studying the details of the Action equation (6. 10)
let us discuss its local invariance properties in a more
geometric fashion. At this point our model consists of angular variables on links and symmetry operations on sites.
To visualize the symmetry operations, imagine a planar
"reference frame" at each site. They are not oriented
relative to each other in any particular way. Interpret
8„(n) as the relative orientation of the frame atn + p to that
atn. We see now that thedefinitionequation(6. 5) is necessary to achieve a consistent geometric interpretation.
The local gauge invariance of the model is now seen as the
requirement that the orientations of the local frames be
unobservable. This is the discrete version of the interpretation of local gauge invariance first discussed by
C. N. Yang and R. Mills (Yang and Mills, 1954). Inparticular, Eq. (6.9a) is the transformation implied by this geometric interpretation when simultaneous, independent
rotations of the frames at sites n andn + p, are made.
The resemblance of this lattice theory to continuum
electrodynamics has already been noted. Let us make
this connection more suggestive by considering the
Action equation (6. 10) at weak coupling (low temperature), where 8, (n) is expected to be a smooth, slowly
varying field. We can then expand the cosine,
1 —cos
8„,= 1 —[1 —' 8'„, +. . . ] = ' 8' „,
—,
and replace the sum over
(6. 11)
lattice plaquettes by an in—,
which is the Euclidean Action. of electrodynamics.
In
doing this we have identified the lattice va, riable 8„(n)
and the electrodynamic potential A, (r) using Eq. (6. 14),
8, (n) =agA „(r)
(6.16)
where g is a lattice theory's coupling co~tant.
The analysis leading from Eq. (6. 10) to (6.15) is plausible but not foolproof even at low temperature. It is
similar to spin-wave (or Gaussian) analyses of spin
models, and these analyses are, as a matter of fact,
usually quite good. However, one should really study
the lattice theory thoroughly, explore its phase diagram, develop a renormalization group for it, and
prove that the Actions [Eqs. (6. 10) and (6. 15)] lead to
the same physics at large distance if T is sufficiently
small. Later we shall carry out such a program for the
planar spin model, Eq. (6.4), and see that the analogous statements do hold true. Such a thorough study of
Eq. (6.10) has not been done, but several approxima. te
discussions have appeared (Banks et al. , 19'lV), and
they support our naive manipulations.
There is one last ingredient in this discussion we have
not motivated, and that is the appearance of the coupling
constant in Eq. (6. 16). In fact, we have made this identification with an eye toward coupling charged matter
fields to the gauge fields in a sensible, gauge-invariant fashion. To make this clear, recall some facts
about local gauge invariance and relativistic field theories first stressed by J. Schwinger (Schwinger, 1959).
In order to regulate the ultraviolet divergence of quantum electrodynamics,
he considered point-split op-
erators,
4(r+ $)y, 4(r),
where g is a quantum'field
which creates a quantum of
charge e. Under local gauge transformations the fields
transform as
g(r) —exp[ieA(r)]g(r),
tt
(r)
- exp[ —ieA(r)]Tt(r),
so it is easy to see that Eq. (6.17) is not gauge invariant so long as (4 0. Therefore that operator must be
modified if it is to be used in dynamical calculations
which are to be consistent. Schwinger suggested the
operator
~p(r+ g)y
r+4
exp ie
r
in four dimensions.
J
Then Eq. (6. 10) becomes
a
A„dx" g(r),
f92
2
~
Therefore, if we identify
= I/2g
8„,=a gE'
„J
Eq. (6. 13) becomes
Rev. Mod. Phys. , Vol.
53, No. 4, October 1979
(6.13)
and the combination
B.
(6. 14)
(6. 19)
because the line integral restores the desired local
gauge invariance. So, if matter fields P were placed
on a space-time lattice, Eq. (6. 19) would read
T|(r + v)y, exp[iaeA„(r)]g(r),
d4
(6. 16)
A. (r) -A„(r)+ 8.A(r),
tegral
(6. 12)
(6. 17)
(6.20)
aeA„has emerged in a natural way.
Gauge-invariant correlation functions, physical
interpretations, and phase diagrams
The partition function of Abelian lattice gauge theory
ls
John B. Kogut: Lattice gaoge theory and spin systems
d8„x
Z=
exp
rgW
—,
I
I —cos
&
8„-&„&„
(6.21)
We want to understand the phase diagram of the theory.
Many of the lessons we learned from Ising gauge theory
carry over to this discussion. In particular, Elitzur's
theorem can be proved here, and again states that the
local continuous gauge symmetry cannot break down
Therefore the ground-state expectation
spontaneously.
value of cos8 (x) vanishes. So, we turn to the gauge-
invariant correlation function to distinguish ordered
from disordered phases of the theory. Consider a directed contour C and the sum of angular variables
around C,
exp i
c
O„x
(6.22)
The same argument that showed us that g„,(r) is gauge
invariant generalizes to a closed loop of any size and
proves that Eq. (6.22) is locally gauge invariant. Therefore we consider (Wilson, 1974)
exp'
c
~~
t
=
K8~(x) exp(-S) jZ.
exp i
C
(6.23)
Suppose g'» 1. Then we can estimate Eq. (6.23) using
standard high-temperature
expansion methods. For
large loops C, low orders in the expansion give vanishing contributions because
(6.24 a)
Thus any exposed phase factor exp[i&, (x)] within the
integrand Eq. (6.23) produces a zero. However, if all
the phases are canceled then we meet integrals,
FIG. 32. The plaquettes which
fiI.l the interior of the contour
& and produce a nonzero
correlation function.
exp i
8, x
(
=,
A
=exp —ln 4g'
A,
6.26
where A is the area or number of plaquettes making up
the minimal surface determined by C. This high-temperature expansion has a finite radius of convergence,
so if we. computed to higher order Eq. (6.26) would become
(exp
i
c
~
~
=exp
g-2
A,
(6.27)
where f(g ') i's a finite function forg' large enough, and
in an expansion its leading term is In(4g'). Clearly all
calculations differ only in dethese high-temperature
tail from the corresponding discussions for Ising lattice
gauge theory.
Next consider weak coupling. If g'«1, we would
guess that a naive continuum limit could be taken and
the Action could be replaced by the Gaussian approximation, Eq. (6. 15). Let us assume this and use suggestive continuum notation,
exp ig
A ~dx
2ff'
de, (~) =2~,
0
(6.24b)
which give finite contribution.
tial of the Action
exp
k~lkt
dg
&&
f
I:I
1
2g 2
cos
„
=
.
„
exp
1
—
d~xP~„+ ig
c
A„dx
(6.28)
1
cos
e~, [ee
1
„e& e 'e
(6.25)
where Z» means a sum over the links of the plaquette
(r, p, v). If we pick out the n= 1 term for each plaquette
making up the minimal area bordered by the large contour C, we pick up a nonvanishing contribution to Eq.
(6.23) which has the fewest possible powers of 1/~'.
'This is visualized in Fig. 32. We have the estimate
Rev. Mod. Phys. , Vol. 5'I, No. 4, October
x exp
Consider the exponen-
1979
where the vector potential A„ranges from -~ to + ~.
We must face two technical problems before we can
evaluate Eq. (6.28). First, the expression is meaningful
only if a particular gauge is chosen (Abers and Lee,
1973). Since the correlation function is manifestly
gauge invariant, all gauge choices will lead to the same
answer. We shall choose the Feynman gauge below.
Second, the theory is formulated with a lattice cutoff
to regulate potentially divergent self-energy effects,
so four-dimensional lattice propagators will be encountered in the evaluation of Eq. (6.28). Lattice propagators have finite values at the origin and are typically
well approximated by their naive continuum expressions
elsewhere (Spitzer, 1964). Since we want only the dominate spatial dependence of the correlation function for
large loops, we shall approximate the lattice prop-
agator,
John B. Kogut: Lattice gaoge theory and spin systems
692
+, (x)&.(0)) = 6„.~(x),
(6.29)
&'(x)
by
~(x
y) =
~(0)6„,+ ~ (x —y),
(6.30a)
=.
1
1
. . if
&a
~x
~
(6.30b)
.
&'(x) = 0, otherwise
Finally, the functional integrals in Eq. (6.28) can be
done by inspection since they are all Gaussians,
where
exp ig
„xA„y
=exp ——g'
A„dx~
dx„dy„=exp ——g'
&
x
-y
(6.31)
dx„dy„
effect. Later we shall understand this electrodynamic
calculation in very simple terms.
To evaluate Eq. (6.31) explicitly, choose a rectangular contour as displayed already in Fig. 2V. It is
convenient to separate the calculation into two distinct
parts as shown in Fig. 34. In Fig. 34(a) one of the four
edges of the contour interacts with itself, while in Fig.
34(b) two different edges interact. Neighboring edges
do not interact because in that case dx dy =O. The
first graph gives
Since the explicit evaluation of the integrals here will
be rather tedious for electrodynamics, let us first
understand the general features of the calculations and
the types of answers expected. First observe that Eq.
(6.31) can be neatly visualized. It states that the line
element dx interacts with the line element dy through
the exchange of a virtual particle described by the lattice propagator &(x), as shown in Fig. 33. Therefore
the dependence of the correlation function on the dimensions of the loop reflects the long-distance character of the propagator. Suppose first that &(x) is
short ranged perhaps a Yukawa potential with a
screening distance p. '. Then we obtain significant contributions to Eq. (6.31) only when dx and dy„are within
this distance. Therefore the correlation function satisfies the Perimeter law. In electrodynamics &(x) falls
off only as a power of ~x ~, so there will be additional
dependence on the contour besides the simple perimeter
—
&'x-ydx
dy
=
dy
~
2
dx—
, [T/a - In(T/a)],
(6.32)
and the second graph produces
I
dx~dy~
=—
dx
dy.
. —y
R +(x
.
2 7T
= ——,
tan
~2
The results for the other edges are obtained from these
calculations by inspection. We shall see that contours
which are much longer in the temporal direction than the
spatial direction have the simplest physical interpretation. Therefore we specialize to the case T»R, simplify Eqs. (6.32) and (6.33), and collect all the terms,
(I/~) T/R —4/~' ln(R/a),
(6.34)
where
P=2(T+R) is
the perimeter
of C. So, the final
result is
R
—
'(T/R) — —ln(l + T'/R')
2
(exp
ig
&„dx
2
= exp ——g'g&+
2
+,
2g
(6.33)
g T
——
2mB
ln(R/a)I,
(6. 35)
where c is a constant depending on the lattice propagator at the origin. We see that the long-range character of the massless propagator &(x) has generated
additional long-range effects in addition to the perimeter law. We shall see below that Eq. (6.35) is just
Coulomb's law in disguise.
Compare Eqs. (6. 35) and (6.26) and note that the correlation function has qualitatively different behavior at
weak and strong coupling. Although the analysis of the
weak coupling limit was rather naive, this result strongly
suggests that Abelian lattice gauge theory is a two-phase
FIG. 33. The loop integral
correlation function in weak
coupling.
FIG. 34. Calculating the loop integral, for a rectangular
tour.
Rev. Mod. Phys. , Vol.
53, No. 4, October 1979
con-
693
John B. Kogut: Lattice gauge theory and spin systems
system.
The gauge-invariant correlation function plays a major
role in our understanding of the qualitative features of
many lattice gauge theories. We should relate it to some
In fact, it measures the force
physical m'easurement.
lau between static charged quanta which can be placed
into the system. To see this, recall that one couples
external currents ~„ to the electromagnetic field by
adding a term,
e
A.
„(x)J', (x)d's,
(6.36)
to the theory's Action. Since charge is conserved, we
can take a closed current loop as a interesting, nontrivial external current. Denote the current loop C.
is unity on the directed loop and zero elsewhere. Choose
the contour to be a rectangle with T»A as in Fig. 27
again. 'Then if we consider a fixed-& slice of the system we see that at x=0, ~p=- 1, while at. x =A, Jp +1.
Therefore, from the transfer matrix point of view, we
have a system with a static charge at x =A and its antiparticle partner at x = 0. The expectation value of the
loop integral can then be interpreted as the ratio of the
partition function for the system, including the external
charges Z(4), to that in which they are omitted, Z(0),
exp
+~dx~
Le
=Z
eJ
Z 0
(6. 37)
If we call P(J) the free energy of the system with the
charges, we have
(exp
ie
c
&„dx„=exp
—F
J
—F 0
(6.s8)
where a factor of kT has been absorbed into 7 [cf. Eq.
(2.6)j. Finally, recall from our discussion of the transfer matrix that free energy density can be identified
with the ground-state energy density of a Hamiltonian
description of the system, Eq. (3.48). In particular,
and appealing to the extensive character
taking
of the free energy, we have
T-
(6.s9)
where the constant of proportionality is the energy difference between the ground state of the Hamiltonian
with the charges included and with them omitted. Since
the charges are static, this energy difference is pure
potential,
s(~) s (o) = v(R) T .
(6.4o)
Therefore Eq. (6. 38) becomes
T,
=exp —V
&„dx
(6.41)
c
so the potential energy required to hold the static
charges a distance R apart is (Wilson, 1974)
exp ie
(
—
p(R)= —lim
ix xxp —ie A„dx
(6.42)
T-" T
c
This is a beautiful connection between the partition function and transfer matrix approaches.
Now we can interpret our calculations of the loop integral correlation function. At strong coupling we have
Rev. Mod. Phys. , Vol. 5$, No. 4, October
)).
t979
the
"area law, " which means
v(R) —iR
f
.
(6.4s)
Thus it would require an infinite amount of energy to
separate the charged sources. 'This result is called
"quark confinement" for obvious reasons. Since confinement occurs so naturally in lattice gauge theories
and since strong coupling problems are handled easily
in this framework, lattice gauge theories give an unconventional perspective to field theory. Next, note that
the "perimeter law" produces a quark potential,
V(R) —const
(6.44)
indicates that charged quanta could easily be
free in such a theory. The forces of the undergauge theory are short ranged in this case. Fiour free electrodynamic calculation Eq. (6.35)
produces a potential,
which
pulled
lying
nally,
V(R) —const. —(e'/2)(I/R
},
(6.45)
which reproduces Coulomb's law f Thus the manipulations leading to Eq. (6.35) constitute "cracking a nut
It is amusing that lattice physwith a sledgehammer.
ics makes strong coupling problems and confinement
appear natural, while weak coupling and free charges
appear peculiar and difficult. The fact that both problems can be handled within a single formalism is, however, quite significant.
Before leaving this topic it is interesting to consider
two-dimensional Abelian lattice gauge theory in some
detail. In this case there will not be a phase transition separating weak and strong coupling. There are
several ways to see this result. For example, if we
repeated the weak coupling calculation Eq. (6.31), we
would obtain an area law, since the two-dimensional
massless propagator does not. fall off with increasing
x. Alternatively we could follow the same line of reasoning which showed that two-dimensional Ising gauge
theory is equivalent to the one-dimensional Ising model.
This would lead to the correspondence
"
Two-dimensional
One-dimensional
(6.46)
planar spin
Abelian gauge
E.
Since the one-dimensional planar spin model is disordered for all nonzero temperatures, the gauge theory
inherits the area law for all nonzero couplings. Let us
pursue this observation in detail and calculate the loop
integral correlation function exactly (Balian et al. ,
1975). Choose the temporal gauge so that
—1 for p, in the T direction. In other words,
exp[i8„(x)]=
e, (n) =o,
(6.47)
is chosen to label representative spin configurations.
The correlation function becomes
exp i
&„=
x exp
d0, n
p
g
gg
where
PV
P=1/2g'. If we choose
xoxp
+i
gc
p
I
the contour to be
X,
„
(6.48)
a rec-
John B. Kogut: Lattice gauge theory and spin systems
tangle, then there are phase factors just on its horizontal (spatial) links. Equation (6.48) can be written in
a manifestly gauge-invariant form by transforming to
(n). Note that in the temporal
plaquette variables
e,
gauge
e, (~, x) =
g e., (~, x).
(6.49)
Therefore the loop integral which appears in Eq. (6.48)
can be written as
n=
iP}
C
&,
(6. 5o)
n
j
where (P, is the set of plaquettes enclosed within C.
Equation (6. 50) is a lattice version of Stokes' law. Using
Eq. (6.49) t:o change integration variables, we have
This example shows the utility of plaquette variables
the correlation function [Eq. (6.51)] in two
dimensions. It is natura1 to ask why this elegant analysis does not generalize to higher dimensions. The
point is that plaquette variables can only be chosen as
independent integration variables in two dimensions.
o see this, let us expose a nontrivial constraintamong
cube.
these variables. Consider a three-dimensional
Label the links comprising each face as shown in Fig.
35. There is a plaquette variable O„„associated with
each of these oriented squares. The orientations have
been chosen so that the sum of all six H„„corresponding
to the six faces of cube sum to zero,
in computing
6)
f aces
„=0.
Such constraints
,
[P,)
de„(n)exp
0
icos& +A+8„„}
Ps
C. The quantum
„... d&„„n
(P, f
cosH
exp
n, wv
„
after cancelling some common factors from the numerato'r and denominator.
Note that Eq. (6.51) is explicitly
gauge invariant. In addition, it expresses the correlation function as the product of independent integrations
over decoupled plaquettes,
r
exp( p cose
de
f
0
„+&e& „)
(6. 52)
de„„exp(p cose~„)
where & is the number of enclosed plaquettes. The integrals in Eq. (6.62) are just Bessel functions of imaginary argument,
8„= I,
(6.58)
Io
which gives us the expected area law for all coupling.
Consider Eq. (6.58) in the limiting cases of strong and
weak coupling. For g'» 1 where p= 1/2g'«1,
.
I, (P)/I. (P) = ' P = 1/4g',
-
Hamiltonian
Choosingg
8
(6.54)
= exp —ln 4g
(6.55)
«1,
(6.56)
so
exp i
~~
=exp ln
and quark
S=P,
P [1
come„(n)]
Pgcose, , (n),
ny
(6. 59)
i,k
where spatial links are labeled with Latin indices and
the tempora1 direction & with the index 0. As usual we
choose the gauge e, (n) =0. Then ~-independent gauge
transformations are still manifest local symmetries of
the system, and they are constructed as in our parallel
discussion of the Ising gauge theories. We shall obtain
an operator formalism for these symmetry operations
below.
To obtain the Hamiltonian, note that the f irst term of
Eq. (6.59) simplifies because
e„(n) = e (n+T) —e (n).
(6.6o)
We know from the general discussion in Sec. III.C that
P, will be forced to infinity in the T-continuum limit.
This will force eo, (n) to be small and slowly varying.
We can, therefore, make the approximation
1 —cos~» = p
~»,
(6.61)
where a, denotes the lattice spacing in the 7 direction.
Sums over the lattice can be replaced by integrals over
the continuum & axes and sums over the spatial lattice,
f
(6.62)
g~ k
1-g
= exp(-g'A)
.
(6.57)
En summary,
the two-dimensional model confines for
all coupling g. The strength of the interquark potential
is a smooth function of g which varies between g' ~R
a.t weak coupling and ln(4g') ~R at strong coupling.
~
~
Rev. Mod. Phys. , Vol.
formulation
Much of the physics of Abelian lattice gauge theory
becomes more accessible using the 7-continuum Hamiltonian approach. The Hamiltonian is obtained by
considering the Action with anisotropic couplings,
soq
exp i
of flux continuity
confinement
(6.51)
exp i
are manifestations
(Gauss' law).
ngtjv
expi
exp i
(6.58)
51, No. 4, October 1979
FIG. 35. Oriented links on a
three-dimensional cube.
John B. Kogut: Lattice gauge theory and spin systems
Now the Action
L,. (n)
with a conventional variable of electrodynamics.
From earlier discussions we have the identification
becomes
S — d7 —P, a,
~
&~
&, n
1
p
Pease, .
n, ik
(z, n)}.
(6.63)
'
So, to obtain a sensible limit, P, must scale as a, and
will
P must scale as a, . The constant of proportionality
be identified with the coupling constant g' which is held
f ixed~
p, =g'/a, -~,
(6.64)
p =a, /g'- o.
The identification of g' with the electrodynamic coupling
constant will become clear later in this section. Finally,
to set up a proper Hamiltonian, we must setupaHilbert
space for each fixed-& surface. There is aquantum field
8„(n) and its conjugate momentum density L~(n). Their
commutation relations are postulated to be
[I., (n), 8„(n')] = i 5,, 5, ,
(6.65)
Inspecting Eq. (6.63) and using our experience gained
from Sec. III. C, we identify the Hamiltonian
aH =
where
"a" is
Q
'g'L', (n) —(1/g')
—,
Q cos8,, (n),
(6.66)
"a" is
the spatial lattice spacing. The factor of
inserted into Eq. (6.66) so that H has the correct
units.
We want to understand the formal and physical content
of these last two equations. First consider 7-independent local gauge transformations.
Such a transformation should add a common angle to all the angular
variables associated with links originating on the site
n. An operator which does this is
G„(n) =expIig&;(n)x};
(6.6v)
(6.v2)
8, (n) = agA„(n) .
This allows Eq. (6.65) to be written in a suggestive
form,
.
(g'/a) [L, (n), &,. (n')] = i 5, , (1/a') &. . .
(6.va)
If we identify a '6„,, as the discrete form of the Dirac
delta function and recall the continuum theory's comrelation,
mutation
[E,. (r), A, (r')] =i6, , 5(r —r'),
(n)
.
(6.v5)
This teaches us an important fact about the lattice theory. Since L;(n) is an angular momentum operator,
i. e. , it is conjugate to the angular variable 8~(n), its
spectrum is discrete,
L, , (n)
= 0, + 1, + 2,
... .
(6.va)
Therefore Eq. (6.75) implies that the electric flux on
a link, a'E,. (n), is quantized in units of the charge g.
Electric flux cannot subdivide into arbitrarily small
units on individual links. The origin of this aspect of
the lattice theory is the fact that agA„(n) is an angular
variable. If we had formulated the theory without this
condition, the quantization of electric flux would not have
occurred. We take this path, however, because it is
the most natural one in non-Abelian lattice gauge theories where the underlying gauge group is compact.
This point will be discussed again in a later section.
If we write the Hamiltonian using Eq. (6.75) we find
H = a'
g
~
E„'(n) —(1/g'a)
n, k
To verify this, note that Eq. (6.65) allows us to interpret the L, (n) as planar angular momentum operators.
Thus Eq. (6.67) is a rotation operator and induces the
change 8;(n)-8, (n)+y for all links originating at n. A
gauge transformation which acts at all the sites and
induces spatially dependent rotations through an arbitrary angle y(n) is
p(x) = exp
l. , (n)x(n)}.
)Q
n, j
(6.66)
8,.„=a'gB, (ij k cyclic).
Then in a smooth, classical continuum limit,
(6.69)
which is an ordinary &-independent gauge transformation. The Hamiltonian is, of course, locally gauge
invariant,
G(q)HG-'(q) =H
.
(6. 70)
(6.vl)
Next we should like to understand the commutation
relation Eq. (6.65) in more detail. In particular, we
should associate the link angular momentum variable
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
=a'
(6. vv)
gij
n,
'
(1 ——, 8',.„)
'+,' 11 + const.
—.
(6. 76)
(6.v9)
,
and conventional
Now
In addition, Elitzur's theorem assures us that the
physical space of states is also locally gauge invariant,
G(X)~&= ~&.
cos 8, ~ = —(1/g'a)
ga
=8, +& y,
g cos8„(n) .
So, the first term is just the simplest lattice form of
therefore, be a
2 J d'xE'. The second term should,
lattice form of ~ J d'xB'. To see this, recall that in the
continuum eleetrodynamies B,- ='c, j~ ~j&~. Therefore we
should make the ident. ifieation
It is easy to check that
G(y)8 (n)G '(y) =8„(n) -X(n)+y(n+k)
electric field, we can
where E(x) is the conventional
identify
E, (n) = (g/a')L,
(6.v4)
electrodynamics is retrieved.
consider the phase diagram of Abelian lattice
Hamiltonian.
Place
static charges +g a distance & apart and consider their
potential V(R). We must first introduce some formalism to describe these charges. 'This will be done by
inventing an angular variable 8(n) which resides on sites.
There will be a momentum conjugate to 8(n), L(n), with
gauge theory using the quantum
the property
[I.(n),
8(n
)]=i6„„.
(6.ao)
John B. Kogut: Lattice gauge theory and spin systems
We want to make a formalism in which a charge g is a
source of g units of electric Qux in accord with Gauss'
law. Therefore matter fields will be described byphase
variables
exp[~ ie (n)),
+L
n,
mill
(6.82)
so that Eq. (6.68) is generalized
G &y) = exp
to
&n)t,
6
Sg 2
The value in this construction
G( ) 8+&&&&a) G 1(x)
&n)}.
(6.88)
lies in the result,
++ax&n& e+&9&n&
(6.84)
which should be compared with Eq. (6. 18), the gauge
transformation properties of an ordinary charged field.
Therefore Eq. (6.84) implies that exp[+i9(n)I carries
+g units of charge.
Next we need an operator to produce a state with
static charges +g. The possibility
exp]- i0(0)j exp fi9 (R)}.,
(6. 86)
comes to mind, but it is not locally gauge invariant.
To remedy this problem, we follow Schwinger and place
a lattice version of exp(ig Jo A &fx) into Eq. (6.85),
'"+ sxpIi Q 9,. &n)} e"'
C
(6.86)
where C is any lattice contour which runs from 0 to R.
It is interesting to interpret Oc(0, R) physically. Since
[L (n)
e+&&&&
flux tube.
(6.90)
&n')]
+6 6
erie&
&&&
i
(6.87)
each operator on the contour C of Eq. (6.86) raises the
on that link by one unit. In other words,
eigenvalue of
the electric flux passing through that link has been increased bye units. Therefore the operator Oc(0, R)
produces a state with the correct amount of flux to satisfy Gauss' law. This is another way of viewing its
property of local gauge invariance.
It is also interesting to determine the cha, racteristics
of that contour C which leads to a state of minimum energy. This question will certainly depend on the value
of the coupling consta. ntg. It lies at the heart of the
quark confinement question. First suppose that g'» 1.
Then the electric term in ~ dominates, so we can imagine treating it as the large, unperturbed piece of the
Hamiltonian and account for magnetic effects in perturbation theory. The zeroth order Hamiltonian
I,
aa, = ,' g' QL,'(n),
Since each link of the contour raises the value of I-k to
unity, that contour of shortest length gives the smallest
contribution to the unperturbed Hamiltonian.
This means
that the flux travels directly between the charges in a
tube, as pictured in Fig. 36, and the energy of the state
grows linearly with R,
(6.91)
&Q c,. &n)v&n) +z Q L
9~&0, R) =8
A confining
(6.81)
and the generator of local gauge transformations
now consist of two pieces,
IJ n
FIG. 36.
—
with our earlier analysis.
If g' were not large, then the magnetic effects in the
Hamiltonian would become important. If we do perturbation t.heory around a&o, it is clear that the lowest
energy state of the +g charges will have its flux occupying paths more complicated than that shown in Fig.
36. As g' is decreased to a critical value we expect
the coefficient of the linear potential Eq. (6.91) to vanish
abruptly. Something like this must occur because at
small coupling the Hamiltonian should be well approximated by ~ d'x(E'+B'), as discussed in Eq. (6.79),
and then Coulomb's law holds. In this region the flux
between charges spreads out into the characteristic
dipole pattern. In the lattice theory, where flux is
quantized, this presumably means that very random
complicated contours like that shown in Fig. 37 lead to
lower energy states than the flux tube. Strong coupling
expansions (Kogut et al. , 1976) and partition function
analyses (Banks et al. , 1977) support these views, but
really precise, convincing work is lacking.
in agreement
f
VII. THE PLANAR HEISENBERG MODEL
0I
M
IN TWO
ENSIGNS
A. Introductory comments and motivation
We have remarked earlier that there are important
similarities between lattice gauge theories and spin
systems. Two-dimensional lattice gauge theories are
equivalent to one-dimensional spin systems. In addition, four-dimensional gauge systems share many
properties in common with two-dimensional spin systems. These similarities, in fact, motivated Wegner
to construct the Ising gauge system (Wegner, 1971).
%e shall pursue this point by studying the two-dimensional planar model in detail. The same type of analysis has been partially carried out for Abelian lattice
gauge theory in four dimensions (Banks et al. , 1977).
(6.88)
is also trivially soluble. In the absence of static charges
its ground state must have the property
L (n) I0) =0,
(6.89)
for all links. In other words, the strong coupling ground
state is an eigenstate of electric Qux with eigenvalue
zero. Now consider the state
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
FIG. 37. Important flux configurations
at weak coupling.
697
John B. Kogut: Lattice gauge theory and spin systems
I.et us recall some facts about the planar model. It
undergoes a phase transition without the appearance
of a spontaneous magnetization.
Its low-temperature
phase contains massless spin waves. The periodicity of
its angular variable 9(n) is irrelevant in this phase. Its
high-temperature
phase is completely disordered, and
the angular character of 9(n) is important here. All of
these properties have correspondences in the Abelian
The analogy runs even
gauge theory in four dimensions.
deeper. In an approximate renormalization group
scheme the phase transition of the planar model is driven
vorby the appearance of topological singularities
tices in the field 9(n). These vortices condense for
T &T, and disorder the spin-spin correlation function.
Similar analyses for four-dimensional Abelian gauge
theories suggest. that topological singularities also drive
them into a disordered phase. These vortices are
closely related to magnetic monopoles (Banks, et. al. ,
1977). The condensation phenomenon leads to quark
confinement in the strongly coupled phase, since electric flux cannot easily penetrate such a medium. These
points wiD be discussed further in a later section.
The idea that condensation of field configurations
having topological significance controls the phase diagrams of gauge theories appears to be more general
than the specific examples we shall study here (t' Hooft,
1978). However, the planar model can be analyzed in
detail, so it is a good place to learn about this phenomenon.
Much of our discussion will be rather technical. Some
intuitive, physical ideas about the planar model will be
discussed in the next section (Kosterlitz and Thouless,
1973). The reader is advised to refer to various review
articles on two-dimensional systems (Kosterlitz and
Thouless, 197V) for more extensive physical discuss lons.
—
—
(J/kT)'"'. Therefore
(exp(i(9, —9„)])= (~/ar)'"'
the order
= exp~- In
(v. 4)
in(kT/~)]
I
Therefore the correlation function falls off exponentially in the distance between the spins for T sufficiently high.
Next consider the correlation function at low temperature. In this case the absence of significant thermal fluctuations suggests that &„ varies slowly and
smoothly throughout the system. Then the cosine in
Eq. (7. 1) can be expanded and only the quadratic term
need be accounted for,
'J
~~
S=-,
'.
n
(v. 5)
n8 2
Then the correlation function becomes,
J
(exp/iftt, —ltd)=—
dg
e'""'"'
xexp ——
4g
2
g,
(7.6)
-
8.
simply because it involves only
Gaussian integrals. If 4(n) is the lattice propagator for
a massless field, then
(7.7)
&exp(i(9, —9„)j)—exp[(kT/&)&(n)] .
which can be evaluated
For large In the lattice propagator
mated by the continuum propagator,
I
is well approxi(v. 8)
which grows slowly at large distances.
Eq. (7. V), we have
(exp'(9.
A'&/2'
9„))=
Substituting
J
into
(v. 9)
The physical picture of Kosterlitz and Thouless
The Action of the two-dimensional
planar model was
recorded earlier,
S=J
p
&nm
cos(9„—9 )=8,
I
Zg
&nm
&
[e"88 8 '+h. c.
],
(V. 1)
&
where (nm) indicates a sum over nearest-neighbor spins
on a two-dimensional
square lattice. In parallel with
our discussion of the Abelian gauge theory, we shall
examine the spin-spin correlation function at high and
low T and argue that the system has two phases. At
high T the correlation function,
(ei88 e-i8„)
d9 ex(80-8„)
',
m
xexp —
J
cos 0„—8
&nm
2lr
d8 =2m,
0
d8
0
(V. 2)
&
can be estimated using a high-temperature
Since
2r
This result teaches us several facts. First, it shows
that the planar model never magnetizes. This follows
the expectation value of the product
because as In
of two spins should approach the product of their expectation values,
e™
=0,
expansion.
But Eq. (7.9) falls to zero as In
(e'8') = 0,
I
—,so
(V. IOa)
(v. lob)
identically. This shows that our approximate analysis
is in accord with rigorous theorems (Mermin and
Wagner, 1966) which prove that continuous global symmetries cannot break down spontaneously in systems
Secwith nearest-neighbor
coupling in two dimensions.
ond, Eq. (7.9) shows that the theory has a line of critical points for T sufficiently small. Recall from the
discussion of Sec. II that at the critical temperature of
a system the spin-spin correlation function is expected
to be power behaved,
(7.8)
our usual arguments show that the first nonzero term
contributing to the high-T expansion of Eq. (V. 2) is of
Rev. Mod. Phys. , Vol. 5't, No. 4, October 't979
-~
(s(0) s(n))—
ln
(7. 11a)
I
where g is a standard
critical index. For the planar
John B. Kogut: Lattice gauge theory and spin systems
treated in the spin-wave approximation
model,
[Eq.
(7.5)],
q = kT/2v
J,
(7. lib)
FIG. 38.
A
vertex.
a fixed line. In the language of Sec. II this
is a violation of the most naive universality criteria.
and we have
Although this analysis of the low-T region of the model is rather informal and is at the same level as our
discussion of the Abelian gauge theory at low T, it suggests that the planar model has two phases. What then
is the nature of the system's phase transition? An
intriguing answer was provided by the seminal work of
J. M. Kosterlitz and D. J. Thouless (Kosterlitz and
Thouless, 1973). They suggested that the periodicity of
the variable 6„allowed for singular spin configurations vortices
to appear in the system at sufficiently
and that they disorder the spinhigh temperatures,
spin correlation function. They considered the. Gaussian
approximation Eq. (7.5) to the model, but they supplemented it with a lattice cutoff and retained the periodicity of its Action. That Action leads to the equation
of motion
—
—
V
8=0
(0&8&2m)
.
(7. 12)
These authors also pointed out that solutions to Eq.
(7.12) could be labeled by their winding number,
VO
-dl = 2wq, q = 0, + 1, +2, . . .
~
~
~
~
.
~
(7. 13)
In the spin-wave analysis leading to Eq. (7.9) we ignored
the periodicity problem, so we effectively only dealt
with the q = 0 sector. A spin configuration having q = 1
is shown in Fig. 38. It is clear that such a configuration
disorders the system significantly, since it stirs the
spins over the entire two-dimensional plane. There is
a famous plausibility a, rgument (Kosterlitz and Thouless,
1973) suggesting that vortices are irrelevant at low T
but drive a phase transition at a moderate T,. For T
~ T, it also suggests that the ground state of the system is a vortex condensate. The argument begins by
evaluating the Action of a vortex (Fig. 38) using Eq.
(7.4),
—v& ln(R/a),
S=
(7. i4)
where R is the linear dimension of the two-dimensional
world and a is the lattice spacing. Thus the vortex
Action diverges as R —~. In addition, the fact that S
reminds us that a vortex is a singular
diverges as
solution to the continuum version of Laplace's equation.
One might think that since S diverges as B —~ vortex
configurations would be irrelevant to the thermodynamics of the planar model. However, to estimate their
importance, consider the free energy of a vortex,
a-0
I' = Action —Temperature
& Entropy.
(7. i5)
F = (&J
2kT) In(R/a)
—
C. The planar model in the periodic Gaussian
approximation
We wish to make the physical picture of the phases
of the planar model quantitative. To do this we shall
introduce a slightly simpler model, the periodic Gaussin planar model, which shares the same symmetries
with the Action equation (7.1). To motivate the new
model note that the periodic character of the Action
equation (7. 1) permits us to write a Fourier series for
each bond,
exp( —p[1 —cos{8„-8 )]] =
51, No. 4, October 1979
g exp[il(8„—
8
g
~~oo
)]I, (p),
(7. iS)
where I, (P) is the Bessei function of imaginary argument. If P were large (low temperature), I, (P) would
be well approximated by a Gaussian and Eq. (7. 19) would
become
exp( —P[1 —cos(8„—8 )]$= (I/v'2vP
)
g exp[il(8„—
8
g~~oo
xexp(- I'/2P)
configuration. Since a vortex is completely specified
by the location of its origin, its multiplicity is just the
number of sites of the system,
Rev. Mod. Phys. , Vol.
(7. 1S)
where + vanishes. Therefore, for all temperatures
equal to and greater than T„we expect vortices to
blend into the ground state vortex condensation.
We can now summarize the Kosterlitz-'Thouless
picture of the phases of the model. At low temperature
spin waves exhaust the relevant spin configurations of
the theory. Spin-spin correlation functions fall off
slowly with distance. Free vortices do not exist, but
bound states of vortex-antivortex pairs can occur. They
do not disorder the system significantly since they affect spins only over small regions. As the temperature is raised, the size of the vortex-antivortex bound
states grows until T, is reached where it diverges.
Then free vortices exist and the ground state is a vortex condensate of indefinite global vorticity. The system is disordered and the spin-spin correlation function falls exponentially.
the entropy of a free vortex. The
entropy is just the logarithm of the multiplicity of the
Therefore
(7. 17)
T, = vJ/2k,
We must estimate
Entropy = k In(R/a)'.
.
The competition between the logarithms of Eqs. (7. 14)
and (7.16) leads us to the interesting conclusion that as
T is increased one reaches a point
)]
.
(7.20a)
side of this expression has the
same essential ingredients as the planar model based
on the cosine interaction. It has the global Abelian
Note that the right-hand
John B. Kogut: Lattice gauge theory and spin systems
and it preserves the periodicity in the &„
Therefore, although it is numerically close
to the original planar model only for low temperatures,
we can consider the right-hand side of Eq. (V. 20a) at
all temperatures and study its phases. Since it shares
the same general characteristics with the planar model,
the two models should have essentially identical--phase
diagrams. It is called the periodic Gaussian model
(Villain, 19V5; R. Savit, 1977). Of course, the real
reason we study it is that it can be analyzed elegantly,
while the original cosine model has only been studied
symmetry
variables.
numerically.
periodic Gaussian model represent
the physics of the planar model'P If we accepted the
strongest universality argument, then we would believe
that their critical behaviors should be identical, since
the models share the same symmetries.
However,
critical lines and nonuniversal behavior are common
in two-dimensional physics, so this argument is not
convincing. We shall see that in the periodic Gaussian
approximation spin waves and vortices do not interact.
In the planar model there are more severe nonlinearities
which could effect. the critical indices. It is not known,
however, if this really happens. This and related points
are subjects of active research. Experiments may, in
fact, resolve these uncertainties. As will be discussed
in the next section, the approximate renormalization
group calculations of the periodic Gaussian model
(Kos'terlltz 1974) predict that. the critical index 'g governing the spin-spin correlation function is 1/4 at the
critical temperature. A low-energy theorem (Nelson
and Kosterlitz, 197V) relates it to the critical behavior
of superfluid helium films, and the data (Webster et al. ,
1979) support this prediction. It would be interesting
if other predictions of the theory could be confronted
with experiment.
Let us develop the formalism of the periodic Gaussian model. We shall see that it is possible to separate the spin waves of the model from its vortices
We shall also
without any additional approximations.
see that the vortex sector of the model is equivalent
Coulomb gas. This point will
to the two-dimensional
shed light on the original spin system, since the phases
of the Coulomb gas can be understood physically. To
begin, write Eq. (V. 20) with more elegant notation,
How well does the
exp(- P[1 —cos(A„8(r}]j—
1
4211'p
gg~
g
(r j -~
exp(in &„8)
xexp(- n'/2P)
where
r
(7.20b)
lattice and n„(r) is
into the partition function,
labels the two-dimensional
a vector field. Substituting
x
'
g
exp(in
b.„8)exp(-
n„(r)/2P),
(7.21)
tip ~&~~
where we have dropped an overall multiplicative constant. The integrals over 8(r) are now trivial. Some
thought shows that each integral generates a constraint,
(7.22)
~.&. (r) = O.
Now,
Rev. Mod. Phys. , Vol.
(„) 0 exp —R~
ny, &r}~-'o
0
t
(7.23)
2
It is best to solve the constraint explicitly,
n, (r) =s, „&„n(r),
(7.24)
where n(r) is a scalar, integer-valued field on the lattice. Equation (7.24) expresses the usual fact that a
divergence-free vector field is a pure curl. The partition function becomes
z=
g
n(r) -~
exp —(1/2P)
P [& n(r)]*j,
(7.25)
t~P
because
n, (r)n„(r) = L~ n(r)]'.
(7.26)
So, we have transformed our original model into that
of a nearest-neighbor coupled integer-valued field subject to the temperature T*=T '. Equation (7.25) is the
"interface roughening" model (Chiu and Weeks, 1976)
of crystal growth, but it is not directly useful to us
because at low temperatures many terms in the sum
over n(r) must be kept in evaluating Eq. (7.25}. Clearly
we would like to replace the integer-valued field n(r)
by an ordinary scalar field @(r) and still preserve the
periodic character of the original model. The necessary manipulation is provided by the Poisson summation formula,
Zgl
)=
g I ~eg(@)e'""',
(7.27)
where g is an arbitrary function. The sum over the
integer variable rn insures that the periodicity of the
original Action is preserved. Applying this identity to
Eq. (V.25) gives
Z=
~OO
x
Qr
„d r
I
exp — 1 2
m&r&-
6 lp (r )
r, l
&„'+2@i
j.
(7.2S)
As we analyze this expression further we shall be able
to identify Q(r) with the spin waves of the original model
and m(r) with its vortices. Equation (7.28} is pleasantly
simple: the vortices act as sources for the spin waves,
which are ordinary massless fields. The identification
of the m(r) with vortex variables is made clearer if we
integrate out the spin waves in Eq. (7.28). This integration is a standard Gaussian, so we obtain without
diff iculty
8 =X.
Q
exp -2mP Qm(~)G(~ —a')m(~')j,
f f
~
(7.29)
.„„d8 y'
Z=
rg
51, No. 4, October 1979
where Z, is the spin-wave contribution produced by the
Gaussian integrations and G(r —r') is the lattice propagator for a massless field. Since spin waves alone do
not drive a phase transition, we shall not discuss Z,
The lattice propagator G(r) satisfies
„.
(7.30)
G(r) =6„0,
&'
where
is a discrete form of the second derivative.
For example, we take
&
700
John B. Kogut: Lattice gauge theory and spin systems
Q2
Q2
(7.31a)
Q2
where
&„'f(~) =f(f. +x)+f(f
(7.38) becomes
Z
-x) —2f(f.).
=Z,
Q
exp ()T'p/2)Q
Then one can easily verify that
G(~) =
I
d ~"
~
27),
~~
(4
2m
~
+eZQ
e t kr
~~
Therefore the variables m(r) interact among themselves
through a logarithmic potential. One can easily check
that vortex spin configurations as sketched in Fig. 38
experience the same force law. vortices of opposite
vorticity experience an attractive logarithmic potential, and vortices of the same vorticity repel one another through a logarithmic potential. Next consider
the self-mass of the field m(f ). This comes from the
f =f' piece of the sum in Eq. (7.29). It is easy to see
from Eq. (7.32) that G(0) is infrared divergent,
(7.34)
In(R/a),
where R is the linear dimension of the two-dimensional
plane. We recognize this as the Action equation (7. 14)
of an isolated vortex. Of course, this infrared problem
means that we should treat Eqs. (7. 14) and (7.32) quite
carefully. It is best to decompose G(r) into two pieces,
one of which is infrared finite and the other is not, ,
(7.35a)
G(~) =G (~)+G(0),
where
'dk "
G(0) =
=Z,
'd '
2m,
1
2v [4 —2
cosk„—2 cosk, ]
into the partition function,
Substituting
Z
(e'""—1)
—
2 cosk, ]
2
cosk„—
[4
2m
„g exp
m
&&exp
(r)
—2m'
The exponential
are "neutral,
—27)'pG(0)
"
.
~
(7. 35b)
~
we have
2
gm(f. )
I
m
x G' ~ —x'
x'
m
singles out those configurations
gm(~) =0.
(7. 36)
which
(7. 37)
Other configurations
to Z. Therefore
do not contribute
we can write
Z=Z,
„g exp
m
(r)
—2e'()g m(r)G'(r —r')m(r')I, (7.28)
where the prime on the sum means "neutral configurations of vortices only. It is enlightening to replace
G' by the explicit formula Eq. (7.33). That asymptotic
form is quite good even for small Ir =1. Then Eq.
"
I
Rev. Mod. Phys. , Vol.
(7.39)
~
I
dk„dk,
2m,
a(r')I,
)
~
Let us examine some of the properties of G(f). For
large x it is well approximated by the continuum massless scalar propagator of two dimensions (Spitzer,
1964)
(7.33)
G(f )= —(I/2)T)ln(f/a) —1/47 (If ~& 1).
(
'm(r)7 (lr —r'l/
(7 32)
~ 2cosk,y ) 'I
—2cosk„~
~
~
G (0) = (I/2m)
m(f-)m(f-r)
yn(g)
(7.31b)
51, No. 4, October 1979
where the primes on the sums over sites mean that the
x =x' terms is omitted. The first term in the exponential can be simplified, since the neutrality condition
Eq. (7.37) implies
0= Q m(f)m(r')
(7.40)
M7
t
'PÃ
K
+
Now)
Z = Z.
„g
exp —(7)'p/2)
m(r)
reZQ
er
(r))m()r —
Qr m'(f. )
~/a)
(r')I.
7f
(7.41)
The first term in the braces gives the chemical potential of each vortex and the second gives the logarithmic
interactions between different ones.
This Coulomb gas representation suggests the character of the phase diagram discussed in the previous
section. If P is very large, then the chemical potential
term in Eq. (7.41) suppresses the vortices very effectively, leaving only the spin waves behind. In fact, the
strong logarithmic potential between vortices suggests
that any vortex-antivortex pairs which might populate
the ground state are tightly bound together. However,
as the temperature is raised vortices are not suppressed significantly. Once P =&/kT becomes of order
unity one would naively expect vortices to become important configurations of the system. Since the interaction between them are long range, they cannot be
thought of simply as free excitations. In fact, they form
a plasma, and a screening length is generated dynamically (Kosterlitz, 1974). It will require a renormalization group arialysis to see this.
D. Renorrnalization
critical region
group analysis and the theory's
The long- range logarithmic
forces between vortices
of the model requires
imply that a detailed understanding
dealing with many-body effects. One tool which was
developed for this type of problem is the renormalization group (Wilson and Kogut, 1974). Before delving
into the details of the periodic Gaussian model, we shall
review the strategy of this approach.
When we write down the local Action of a spin system
or field theory, we see clearly how degrees of freedom
interact over distances of the order of the lattice spacing
a. However, frequently we are interested in the longdistance characteristics of the theory or, equivalently,
its low-energy
content. The original lattice Action does
transparent about these character-
not tell us anything
John B. Kogut: Lattice gauge theory and spin systems
istics of the theory. The trouble is that cooperative
effects occur. Some details of the lattice Action may
be important to the interactions between widely separated degrees of freedom and others may not be. Cooperative, many-body effects determine which elements of the lattice Action are important. To understand the critical region of a theory where the correlation length is huge, we need a description of the
theory formulated directly in terms of its long-distance
characteristics. We might generate such a description
from the original lattice Action by averaging out some
of its short-distance degrees of freedom. This averaging procedure may be difficult to carry out in practice, and the effective description of the system may
be quite complicated if cooperative effects are really
important. 'The renormalization group approach does
the averaging in small steps so that each computation is
tractable. A sequence of effective Actions is then generated by iteration. Large amplification or deamplification effects, which are expected when we begin
with a lattice Action and end with an effective Action
which describes the bulk properties of the theory, are
obtained from the iteration procedure. The aim of such
computations is to uncover the long-distance physics in
the original model. The final effective Action might
not have an appearance similar to the original Action,
but it has the same physical content.
We shall carry out this program for the periodic
Gaussian planar model.
We shall work in momentum
space, using continuum physics methods, and shall
systematically integrate out the high-frequency parts
of the theory. In effect, we shall obtain an effect. ive
Action for the low-frequency (or long-distance) content
of the theory. The phase diagram of the effective Action
will be clear almost by inspection.
To begin, return to Eq. (7.28),
Z=
d
'v
~()O
xZ
exp —(/yp
P (e p)* (P
y
(er)pr(r)Ie.
and y, is an adjustable parameter we shall discuss
further below. The original model is defined by y, = 1.
To be consistent one must equate the lattice propagator
with the pure logarithm,
G'(&)
-- (1/»)»(I~ I/~»
(v. 28)
m(r)-o, ~j
exp (inym'(x) + 2'
—
X(y)
f
''dp(r)g
e)ey
eyri Qm(r)@(r)I,
r
(v. 42)
(V. 41),
y
=y, exp(- m'P/2),
Rev. Mod. Phys. , Vol. 5't, No.
(v. 43}
4, Octaber 1979
(v. 45)
The partition function becomes
Z(y)
=
f;"Idp(r)
r
&
@(~)—0
where y includes the chemical potential observed in
Eq.
=1+2y cos[2)T(t (~)]
= exp (2y cos[2)T(t) (x)]j .
exp —
1
2P
To restore conventional
ry P
Q m*(r)
(x)@(x)]
+~
+2/
cos
23'
(v. 48)
exp —)/y()g(e, p)'
m&r&
1l
m
—1+ exp(lny)(exp[2vi(t (r)] + h. c.)
=
Let us first anticipate some of the effects of integrating
out the high-frequency components of this expression.
Equation (7.41) shows the effects of a complete integration over P(x) while leaving the vortex field unina chemical potential term and long-range
tegrated
interactions of the vortices occurred. It is convenient
to anticipate the generation of a vortex chemical potential when the high-frequency components of the spin
waves are integrated out, by incorporating such a term
into Eq. (7.28) from the start (Jose et a/. , 1977),
(v. 44)
since the —1/4 piece of Eq. (7.33) is already accounted
for in Eq. (7.43). We make these definitions so that we
can compare with published accounts of the resulting
renormalization group (Kosterlitz, 1974). Note that
the Action is now parametrized by two quantities: the
temperature and the chemical potential of a vortex. The
renormalization group analysis will produce effective
Actions having the same form as Eq. (7.42) but with
different parameters: an effective temperature and an
effective chemical potential. Let us anticipate how the
effective chemical potential will behave as a function
of temperature. At low T we expect vortices to be
suppressed. This means that although they can show
up in tightly bound pairs of zero vorticity, they will not
affect the long-distance features of the theory. It is
reasonable, therefore, to expect the effective chemical
potential of a vortex to grow large as more and more
high frequencies are integrated out, if the temperature
is small. Then only the m(x) =0 term of the partition
function would be significant. However, as T is raised
we expect to find a temperature T, above which vortices
populate the ground state. This should be reflected in
the tendency of the effective chemical potential to decrease as more iterations of the renormalization group
are done for T &T,. We should be able to test for these
qualitatively different types of behavior by studying
Z(y) for small y: at low T an initial small value of y
should iterate to zero, while above a critical temperature an initially small y should grow. All of the analysis which follows in this section is aimed at establishing these points.
Note that if y is very small then Eq. (7.42) can be
simplified. The point is that if y «0 only the terms
m(x) =0, + 1 need be kept in the sum over vortices. Then
r, u
m&r)
701
(~)/iso
&(y)
field-theoretic
notation,
rescale
f. .dp(r)=
..
1
2
cos
27T
r
(v. 4v)
702
John B. Kogut: Lattice gauge theory and spin systems
which we recognize as the quantum sine-Gordon theory.
'This .is a convenient form for developing a renormalization group of the theory. We shall work directly in
momentum space and regulate the theory with a momentum cutoff A. We shall then integrate out high frequencies of the theory, i.e. , frequencies near but lower
than l~, and find an effective Action for the lower frequency modes of the theory. This approach (Raby and
Ukawa, 1977) complements the more familiar real space
renormalization group analysis done using the Coulomb
gas representation of the theory (Kosterlitz, 1974).
To begin, define a momentum space cutoff version of
Eq. (7.47). All Euclidean momenta P must have magnitude less than A,
exp
P
d2
h(x)=@ (x) @,. (x)=
&2+
(). &p&~
, e*'"y(p) .
(7. 52)
Next, organize ZA in the same way. The kinetic energy
piece of the Action is easily split,
dp
2
277
d2
(2„).P'4 '(P
d x Q ~, &
+—
(t) ~,
d'x
.
h&h
(7. 53)
-S
Therefore, the partition function becomes
(7.46)
0&p&A
z„=
where &Q(P) denotes a functional integral over all the
momentum components of the field,
s[P 1=
f
d'xI ——@ (x)«'@ (x) —oooe[2x«ZP
0&p&A '
fdx@
eo@(p)expI5
op. jz',
(4 54e)
,
where
Zr—
(x)[j
' &p&A
f
«OP(P)exp . —
2
+p,
(7.49)
d
xoe'5
dxcos 2r
A
and
+h
(7.54b)
(7.50)
Since p, ~y which is assumed to be very small, we shall
evaluate Z' in perturbation theory. Let us introduce the
notation for averages of quantities & in the high-momen-
The parameter p, in Eq. (7.49) is just 2y/a'.
To integrate out the high-frequency components of
range 0&p &A into a low&(I)~(x), split the momentum
frequency part 0&p & A' and a high-frequency part &'
&P &&. Define a field which only has momentum components in the lower slice,
.d .. e""4(P) .
0, . (x) =
part of
Z'=1+ p.
(t)~
slice,
' &p&A
(7.51)
27t')
Then the high-frequency
tum
' &p&A
'Then
will be
'
, —p,
d'x(cos[2wv p (&t&~+h)])„+
h
2
f
&@(p) exp — d xoo
554 (p) exp
2
oj'*
— d'xoe'oj.
2
(7. 55)
Z' becomes
d'xd'y (cos(2m'[(t)~, (x)+h(x)]] cos(2mvp [p~, (y)+h(y)]j)„+. . . ,
(7.56)
where an unimportant overall multiplicative constant has been dropped. The expectation values in Eq. (7.56) are
easy to evaluate, since they are taken with respect to a free, cutoff field. For example,
(cos[2m5(p (p~,
+h)])„=~(exp[2miWp(p~, +h)]+h. c. )„'
= 2 exp (27)
where G„(x) is the free propagator
G„(x) =
&( &
~ (27)')
Since these quantities
A. (x) = exp[-
i'@~, )(exp(2riv'Ph
)) „+h.
for the high-frequency
c. = exp(-
components
'
will occur frequenctly
2m
PG„(0)] cos [2wWPP ~, (x)],
(7.57)
of P~,
(7.56)
below, define
(7.59)
2m'PG„(x)],
so Eq. (7.57) becomes
&cos[2mMP(y~, +h)]&„=A(0) cos[2mMP@~,
(x)].
(7.60)
Consider next the connected part of the third term in Eq. (7.56),
&cos(2w5(P [(t) ~, (x) +h(x)]] cos(27) vP [@~,(y)
+h(y)]])„
—(cos (27) v P [y ~, (x) + h (x)]])„&cos(2w WP[y ~, (y ) + h (y )]))
Calculating as in Eq. (7.57), we find that Eq. (7.61a) equals
„.
Rev. Mod. Phys. , Vol. 5't, No. 4, October 1979
(7.61a}
703
John B. Kogut: Lattice gauge theory and spin systems
' A'(0) [A'(x — — cos 27[ x
P [@~,(x) + (P ~, (y )] + ~ A'(0)[A '(x
y ) 1]
—,
this expression if we consider only that
region of real space-time where it is appreciable. The
propagator G„(x —y) should be non-negligible only for
slicing is done smoothly
~x —y &I/&' if the momentum
enough. Therefore the presence of [A '(x -y) —1] in
Eq. (7.61b) allows us to expand the difference,
We can simplify
z =-,'(x+y)
(V. 62a)
.
(V. 62b)
Eq. (7.6lb) becomes approximately
'A'(0)[A (g) —1] cos[4vvP (t)A (z)]
' 4 'P[gs@,, )]'.
—,
'A'(0)[A-'(g) 1](1 —,
(
Finally, Eqs. (7.57) and (7.63) can be substituted
(7.56). If one defines various constants,
Now
&'
a =
'd'
a3 =
d' A
carries out
substitutes
z
(v. 63)
0„(z)
—1
(v. 64)
—1
4
1+
f
&0(P).
d*zj
o&g
~2 ~2+
e eA (0)
&g+
a
O
8
d g
7 65
0„.(z)]d'z j,
eee[xzdd
and
—v'1+ —,' 7('P p'A'(0)a,
(e, d)=expI-',
(
1 =
d2p,e' ~ —,
),
d&
1
(
)d„{A[z[),
e.*A*(0)e,
where the new parameters
a2-4~
then Eq. (7.65) becomes
f
d*zjZ
(00'),
(v. sv)
are given by
i[, z=A(0)i[, ,
(v. ss)
P' = P/[1+ ,' v'P p, 'A'(0)a,—].
In addition, the change in the overall scale of the partition function means that the free energy density
F''= —InZ'/f d'x has changed,
E = F ' 0 y. 'A'(0)a, .
(7.69)
In conclusion,
Eq. (7.67) provides us with a new parametrization of the original model. The physics at the
point (p, , P) is the same as the physics at (p, ', P'). The
calculation has shown that, when high-frequency fluctuations are integrated out, an effective Action is obRev. Mod. Phys. , Voj. 51, No. 4, October
1979
)
(1/2. ) -~.(~~ ~x)-,
(7. 72)
where the quotes indicate a function obtained from a
smooth momentum space slicing calculation. With the
same procedure the quantity a, becomes
(v. ss)
(j)~. ,
rescale momenta p-p/2,
Z
f
G„(x) —(dJ /J
where we have ignored induced interaction terms which
are higher order in p, than the term pA(0) cos(27[WP@d(, )
retained. If we rescale the field &f&~,
(t) ~,
a„a„and
(7.71)
1
—
eA'(0)e,
exp
(v. vo)
where we have done an angular integration to expose
the Bessel function. Unfortunately,
Eq. (7.71) is not.
good. We have employed a sharp momentum space cutoff [Eq. (7.70)] and found that the propagator in real
space behaves like a Bessel function. But Bessel functions do not fall off rapidly as their argument increases.
Therefore, the naive estimates leading to Eq. (7.63) do
not go through. This flaw can be solved by considering
smooth momentum space slicing (Wilson and Kogut,
1974). We shall not need to execute such a procedure
here, but we need to know that such a calculation can be
done in principle. Then G„(x) is really a short-ranged
function. Let us write Eq. (7.71) in the generic form,
the integrations in Eq. (7.56), and finally
back into Eq. (7.54), then
=—exp
(7.61b)
Now the propagator G„(x) and the constants
a3 wil l simpl ify. For example
into
—1
A
d.
(x) —(t ~, (y)] .
dA&p&A.
A
—,
a, =
d, ,
«1.
where
(=x —y,
cos2vrMP[(g&
tained which differs from the original Action by
(1). wave-function renormalization, Eq. (7.66), (2).
coupling constant renormalization,
Eq. (7.68), (3).
ground-state energy shift, Eq. (7.69). The form of the
effective Action is the same as the original Action only
because of the assumption p,
If p. were not infinitesimal, interaction effects of the form cos[4wvP &f&A, ]
would occur in Z' and could not be ignored.
To understand the implications of Eq. (7.68) it is convenient to consider an infinitesimal high-momentum
slice A'=A. Then Eq. (7.68) will reduce to differential
equations. The high-energy slice will be defined by
~
= 5 sf, (~),
4, (x) —@,(y) —
- y ) —1]
P&
d~~
~»
(7.73a)
where
,
',
2=
dpp3 ~o p
(v. v3b)
is a dimensionless number dependent on the slicing
procedure.
Differential equations for p and P follow. Consider
Eq. (7.68) for the parameter i[, ,
5 p = p, ' —p, = p, [A (0) —1] .
(v. v4)
Since A (0) becomes
A(0) =1 —2m'PG„(0)
= 1 —2~'P (da/A) (I/2~)
for an infinitesimal slice, the differential
P is
dV = —~Pe(«l&).
(v. v5)
equation for
(v. vs)
704
John B. Kogut: Lattice gauge theory and spin systems
equation for P follows in a similar
The differential
fashion,
dP = —2m'o. tu'P'(dA/A')
,
.
(v. '77)
To compare with other derivations of this renormalization group, Eqs. (7.76) and (7.77), we pass to a realspace cutoff Q = A ', so
d p, = —wP p, (da/a),
d(6
= —2m'c(,
p.
(v. v8)
'P'a'da.
Recall from Eq. (7.42) that the natural variables for
the problem are P and y = pa . The differential equation
for y reads
dy =Q
i),
(da/a)a'+ 2 p, ada,
a(~P
.
2)da
{v.v9)
.
Inspecting the equations for dP and dy, we note that
y
=0, p=2/7),
(v. 80)
is a fixed point of the coupled differential equations.
herefore, define
x =mP —2.
(v. 81)
little algebra shows that the quantities x and y satis fy the diffe rential equations,
A
2dQ
=—
2xy
Q
dx = -167' &2y
(v. 82)
=0 consists just of spin waves, as Eq. (7.47) shows'
Therefore, we have rediscovered the fixed line of the
planar model all of these trajectories flow to the positive x axis. The trajectory labeled T = T, is the boundary between those which flow to y =0 and those which do
not. To the left of this boundary the trajectories flow
toward larger and larger y values. Therefore, since
y measures the importance of vortices in the system,
we have indeed discovered that vortices drive the phase
transition of the planar model and give rise to a qualitatively distinct high-temperature
phase. The point
(x, y) = (0, 0) labels the end of the fixed line and is therefore associated with the critical temperature.
To analyze Eq. (7.84) more precisely, recall that in
the original model
(v. 85)
x =wP —2 = (mJ/kT) —2, y exp( —m2J/2kT) .
The critical temperature is then determined by the condition
Q
x
"
becomes
= —2xy 2dQ '
Q
(v. 83)
2dQ
Q
We have worked hard to put the renormalization
group
equations into this form because they have been
thoroughly studied in a related problem (Anderson et al. ,
1970).
We can now understand the character of these equations and their implications for the phase diagram of
the planar model. Note that there is a trivial first integral of Eq. (7.83),
x 2 -y 2 =const.
(7.84)
The trajectories of the differential equation in the (x, y)
plane are, therefore, hyperbolas, as shown in Fig. 39.
First, consider trajectories in the region of the plane
labeled T &T,. They flow into the x axis a, nd stop in
accord with Eq. (7.83). Since any two points on the
same trajectory give the same physics, all trajectories
in this region of the plane define theories which are
equivalent to ones having y = 0. But a model having
Rev. Mod. Phys. , Vol.
group trajectories of the planar
y
2dQ
in the vicinity of the fixed point Eq. (7.80). Finally, by
changing the scale of "Q, the second of these equations
can be simplified. The first equation is unaffected by a
change of scale, since it involves only the scale-invariant differentials dy2/y2 and da/a. So, finally Eq. (7.82)
dx=-y
FIG. 39. The renormalization
model.
—
dP + @dc
= —mP
X
51, No. 4, October 1979
-y =0,
(V.86a)
which becomes
wJ/kT, —2- exp(-7)'J/2kT, ) .
(7.86b)
Therefore, the nonvanishing of the right-hand side gives
a correction to our earlier, naive estimate of T, T,
=7) J/2k. Since the correction behaves as exp(-1/T), it
is due to vortex interactions which were neglected in the
earlier, intuitive discussion. If we expand x -y' about
zero using Eq. (7.85), we can find the constant of Eq.
(7.84),
= —C
(v. 8v)
where c is a positive quantity =2. 1 (Kosterlitz, 1974).
A more detailed analysis of these differential equations allows one to calculate the correlation length in
the critical region. If one accepts the plausible assumption that trajectories labeled by T &T, flow to infinite temperature, then one can show (Kosterlitz, 1974)
((T)exe exPI8(
'
. X/2
.
(7.88)
C
Therefore $(T) diverges with an essential singularity
and the usual power-law hypotheses for critical singularities, as reviewed in Sec. II. A, do not hold in this
model. However, correlation scaling does apply. For
example, dimensional analysis suggests that the free
energy density, measured in units of kT, should behave
John B. Kogut: Lattice gauge theory and spin systems
f/kT- ( (T)
(7.89)
in the critical region, and this result does follow from
the differential equation (7.69). This result shows that
the free energy and all its derivatives are continuous in
the critical region. And finally, the spin-spin correlation function is predicted to behave as
(expi(60 —8„))-
1
(7.90)
at the critical temperature T„and the index rl is 1/4 as
in the Ising model. This last result agrees with experiment (Webster et a/. , 1979).
In summary, this model illustrates how topological
singularities in a lattice spin system can drive a phase
transition. The final conclusion is that the low-temperature phase is given by our naive, smooth-field exit is exhausted by spin waves (aside from
pectations
some finite renormalization effects). The high-temperature phase consists of a vortex condensate and has
short- range spin- spin co rrelations.
It is interesting that aperiodic Gaussian version of Abelian lattice gauge theory in four dimensions can also be defined and analyzed along these lines (Banks et al ., 1977). In
that theory the topological singularities consist of closed
lines of vortex singularities.
It appears that at low T
these vortex singularities are bound together into neutral nets and do not affect the long-distance characteristics of the system substantially.
Then our naive expectation that the theory reduces to the ordinary free electromagnetic field should hold. However, at larger T the
nets of vortex singularities should grow into infinitely
long loops which disorder the system and lead to quark
confinement.
Unfortunately, reliable renormalization
group analyses have not yet been done on the theory, so
although these points are plausible they are not proved.
705
Consider two nearest-neighbor frames, one at site n
and the other at site n+ p, . Their relative orientation
is specified by a rotation matrix which lives on the link
between them,
U„(n)
=exp[iB
(n)],
U„(n+ p) = U (n),
(8.1a)
where
A'
B (n) = 'ag r;—
(8.1b)
Eq.
(n)
and the color index i is summed from 1 to 3. In
(8.1b) we are anticipating the connection of this formalism to ordinary continuum non-Abelian gauge fields.
A gauge transformation will be a local rotation of a
frame of reference. For example if the frames at sites
n and n + g are rotated, the U (n) transforms as
—
VIII. NON-ABELIAN LATTICE THEORIES
A. General formulation of the SU(2} theory
It is not difficult to extend our general considerations
to non-Abelian local symmetries.
The physics here is
much more exciting and, hopefully, is relevant to the
world of strong interactions.
Let us introduce the idea of a local non-Abelian symmetry in a geometric fashion, following the original
ideas of Yang and Mills (Yang and Mills, 1954). Consider a cubic lattice in d-dimensional Euclidean spacetime. Let there be a "frame of reference" at each site.
If the local symmetry group is SU(2), the frame of reference is three dimensional and refers to the internal
symmetry (color) indices of the gauge group. Suppose
that these frames can be oriented arbitrarily from site
to site, so that color cannot be compared at different
points in space-time. For example, the theory might
have colorful quanta which can hop from site to site and
whose color is measured relative to the- local frames of
reference. We want to construct the Action of the
theory so that it is invariant to changes in the orientation of the local color frames of reference. The existence of a colorful gauge field arises from this symmetry principle in a rather natural way (Yang and Mills,
1954) .
Rev. Mod. Phys. , Vol.
61, No. 4, October 1979
x(exp[i ,'T, y -'(n+ p)]]„[U,(n)]„
(8.2a)
where the SU(2) matrix
'v y (n)],
exp[ —i —
(8.2b)
describes the local rotation at site n. One can also
visualize the transformation law [Eq. (8.2a)] in terms of
the quantum-mechanical
spherical top. Then U (r) is
the rotation matrix describing the relation between the
space-fixed coordinate system and the body-fixed coordinate system (Kogut and Susskind, 1975).
Consider the theory with only gauge field degrees of
freedom, U, (n). A locally gauge-invariant Action can be
written down following the motivation discussed for
Ising and Abelian theories,
xU (n+
p.
+ v) U„(n+ v) +h. c.
(8.3)
It is easy to check that the Action incorporates the local symmetry equation (8.2a). In addition, the link variables B (n) enter the Action only through rotation matrices, so their range of variation is naturally compact
(0 &B„(n) &4n). This point should be contrasted with
our discussion of Abelian lattice gauge theory. There
we constructed a theory based on phase variables, and
periodicity in the variable a@A (n) followed. This was
not forced upon us. However, for non-Abelian theories
it is natural to form a lattice version with group elements U (n). Since SU(n) groups are compact, the
bounded character of B (n) follows.
Let's understand Eq. (8.3) in more detail. The first
feature we should establish is that for classical, smooth
fields it reduces to the ordinary Action for non-Abelian
gauge fields. To begin, write Taylor expansions for the
gauge fields,
B„(n+ p) = B„(n) + aB,B„(n)
B (n+ p+ v) = —B (n+ v) =—[B (n)+aB„B
B „(n + v) = -B„(n) .
-
(n)]
(8.4)
Then the matrix product of four U's around a directed
plaquette becomes
John B. Kogut:
U (n) U, (n
+ p. ) U
(n
Lattice gauge theory and spin systems
+ p. + v) U, (n + v)
x
= exp(iB )
Now
exp[i(B„+ aS, B,)]
x exp [—i(B~ + as„B~)]exp( —iB,) .
apply the Baker —Hausdorff formula,
U (n)
U„(n+ p)U (n+
p.
+ v)U, (n+
v)
(8.6)
=exp(i(B
= exp(ia(B
s.„=s.a„—s„a. + ig[a„,a„],
= ~7;.A'. P „ is the standard Yang —Mills field
strength. Note that the commutator [A, , A„] appears
naturally in P „as a consequence of local gauge invariance. For smooth, classical fields we can assume
that a~gP „«1 and simplify Eq. (8.7) further,
where A
~
„+
}
(8.9)
The tr 1 term has no dynamics in it and can be dropped.
vanished when the trace was
The term linear in
taken. Using the commutator
P,
(8.10)
[~, , ]T= 2i s,
we find
(8.11)
Now the Action
~+@+(1/2)[x, yg+
(8.6)
Eq. (8.6)
~ ~
We shall not need the terms denoted by dots in
because in our application they will be multiplied by
many powers of the lattice spacing and will not contribute in the continuum limit. So,
+B, + as B„)—2[B,B„]]exp[ —i(B +B, + aB„B~) —~[B,B,]]
B, —B,B ) —[B,B„P= exp [ia gF~, ]
where we have defined in the last step,
'a"g'7',
tr exp(ia'gP„) = trf1 + ia'g5'„——,
—
=tr ]. —'g4g' try' + - .
y
becomes
(8.12)
So
(8.13a.)
where
(8.13b)
(8.7)
the dramatic difference between Abelian and non-Abelian theories lies in coupling-constant renormalization.
We saw that for T &T„ the effective temperature of the
Howplanar model did not renormalize significantly.
ever, above the critical point, the effective temperature
rose as coarser and coarser lattices were considered.
It was tempting to assume that for T &T, the renormalization group trajectories flowed to an infinite temperature fixed point (Kosterlitz, 1974). In non-Abelian theories perturbation theory calculations show that even for
weak coupling the renormalization group trajectories
flow toward stronger and stronger coupling (t' Hooft,
1972; Politzer, 1973; Gross and Wilczek, 1973). This
suggests that the non-Abelian theories do not have a
distinct weak coupling phase of massless particles and
Coulomb-like forces. Rather, for all coupling the
theory resembles the high-temperature
phase of the
Abelian models, in which a mass gap develops dynamically and the system is disordered.
The property by which the renormalization group trajectories of non-Abelian gauge theories flow toward
strong coupling is called "asymptotic freedom" (t' Hooft,
1972; Politzer, 1973; Gross and Wilczek, 1973). Consider two versions of the theory, one with a space —time
cutoff "ao" and another with space-time cutoff "~. We
can think of go and g as lattice spacings, although the
calculations are done using the continuum formulation
of the model. In order that the physics of the two formulations be identical, their coupling constants must be
"
related,
2
The reader should recognize Eq. (8.13a) as the usual
classical Euclidean Action of pure Yang —Mills fields.
B. Special features of the non-Abelian theory
To discuss the continuum limit of the field theory
equation (8.3) we must treat B (n) as a legitimate fluctuating variable. One might guess, however, that for
weak coupling the classical analysis should be a good
guide to the physics of the theory. This was the case
for the planar model in two dimensions. There we
found that for T &T, the vortices were irrelevant, and
the theory was well described by massless spin waves.
Presumably, an analogous result holds for Abelian lattice gauge theory in four dimensions. However, there
are good reasons to believe that the classical Action is
never a good guide to the non-Abelian models& This is
known to be true for non-Abelian spin models in two dimensions, as we shall discuss later. The reason for
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
2( )
g'0
1 + (Cga/2m') ln(ao/a)
where C is a positive group-theoretic
that as ~-0,
g (a)
—2m/C
ln(ao/a)
—0,
(8.14)
'
constant.
Note
(8.15)
limit (a-0) of the theory is at zero
coupling. Since the interacting theory is weakly coupled
at short distances, detailed comparison between theory
and experiment are possible for processes which are
only sensitive to the short-distance features of strong
interactions. These comparisons have been very successful, and such detailed successes have strengthened
many theorists' belief that non-Abelian gauge fields provide the basic hadronic force.
The long-distance properties of the theory are not
well understood. Equation (8. 14) suggests that the effective coupling must be large when the theory is formulated with a large space-time cutoff. If this is the
so the continuum
John B. Kogut: Lattice gauge theory and spin systems
case, one would expect the theory to have a rich spectrum of bound states. In addition, the force law between static quarks which belong to the fundamental
representation of the gauge group could be confining.
Certainly the strong coupling lattice theory, Eq. (8.3),
has these properties.
Given our experience with simpler theories, we can
discuss the non-Abelian lattice Action easily. Since it
incorporates a local symmetry, Elitzur's theorem applies to it, so the theory cannot have a local order parameter. Wegner's construction of a gauge-invariant
correlation function generalizes to continuous non-Abelian groups. Consider a closed, directed contour C and
the operator
(8.16)
, , exp[iB, (n)].
C
We can study
C
its expectation value,
exp[(B (n)]) = Z(C)/Z,
where
Z(C) =
f,
tip
(8.17a,)
SB )(enxp„( —S)
exp[iB (n)]
(8.17b)
C
P
and S is the lattice Action equation (8.3). Our usual
high-temperature
expansion methods show that for g
large enough the area law applies,
(
exp[rB
C
„(n)])= exp[-B(S ')B],
(8.18)
where A is the area. of a minimal surface enclosed by
The phyC and E'(g ) is a well-defined, finite function.
sical interpretation of this correlation function is the
same as the Abelian theory it gives the force law between heavy, static quarks. Of course now the quarks
carry a non-Abelian charge, color. Assigning the
quarks to the fundamental representation of SU(2) color
means that under local color gauge transformations
they transform as
—
y, (n)
—[exp[ad-,'~, q'(n)]]„q, (n) .
(8.19)
Then to make a bilocal operator such as Z, T[);(n+ ][[)g;(n)
locally gauge invariant, we must follow Schwinger's
construction and form the operator
et B~(n)
(8.20)
So, as a quark hops from site n to n+ p, it must be accompanied by a gauge field rotation matrix on the intervening link. Therefore, if one specifies that the quark
hop around the closed contour Q, the gauge field piece
of the amplitude is clearly Eq. (8.17b). Finally, if a
rectangular closed contour (Fig. 27) is considered, we
obtain the quark-antiquark
potential,
707
In summary, the lattice Action has simple properties
at strong coupling which may be properties of the strong
interactions. The continuum quantum theory has simple
short-distance characteristics which compare well with
experiment. Are these two theories related? It is
hoped that they lie on the two ends of one renormalization group trajectory and that the theory of non-Abelian
gauge fields enjoys asymptotic freedom on fine lattices
and confinement on coarse ones. The complexity of the
theory has prevented a constructive proof of this point,
the Holy Grail of the subject. However, strong coupling
expansions have been used to search for possible phase
transitions in the intermediate coupling region of the
theory, and none were found (Kogut, Sinclair, and Susskind, 1976). It appeared that the only critical point of
the theory resides at @=0, but the analysis was not
strong enough to be convincing. An approximate recursion relation due to A. B. Migdal (Migdal, 1975) also
suggests that the theory resides in just one phase.
Some remarks about his results will be made in a later
section. Much more work (and inspiration) is needed
on this crucial subject.
Throughout these lectures parallels have been drawn
between lattice gauge theories in d dimensions and spin
systems in d/2 dimensions.
Such connections also exist
for non-Abelian theories. First, it is easy to see that
two-dimensional SU(2) lattice gauge theory is equivalent
to the SU(2) &&SU(2) Heisenberg spin system in one dimension. Choose the "temporal gauge" Bo(n) =0. Then
the lattice Action reduces to
S = —(I/2g
)
P tr U„(n)
+ ~) + h. c . ,
(8.23)
(pe r) +X c.
(8.24)
U „'(n
or
S=P
x
—
2C
Ptrp, (n)U,
I
where sites of the lattice are labeled n=(x, ~). Therefore the original Action breaks down into many copies of
a nearest-neighbor coupled one-dimensional spin model.
Since the U matrices transform according to SU(2)
XSU(2), the spin model is the SU(2) XSU(2) Heisenberg
chain. And, since SU(2) &&SU(2) is identical to O(4), we
can visualize Eq. (8.24) as copies of a chain of nearestneighbor coupled four-dimensional unit vectors. To
make this point explicit, note that since U„(n) can be
represented as a 2x2 unitary matrix it can be parame-.
trized in terms of Pauli matrices 7;. ,
U„(n)
=o(n)+sr
where (o', g) are four real fields.
U„(n) be unitary is
o'(n) + w'(n)
(8.25)
w(n)
=1.
So, consider a four-dimensional
link,
The condition that
(8.26)
unit vector on each
(8.21)
by essentially
repeating the arguments which led to Eq.
Then Eq. (8.18) gives the
linear confining potential
(8.27)
(5.42) for the Abelian theory.
(8.22)
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
i7r3
Using the identity
John B. Kogut: Lattice gauge theory and spin systems
708
trU„(n)U„'(n +r) =2[a(n)o{n+ 7') +s(n) s(n+
=2s(n) s(n+
~
T)
effective Action can only have terms of the form
&)]
S'= (I/2g')a s
(8.28)
~
B, s+
(1/2f ') {8 s s„s)'+
~
~ ~
~
.
(8.32)
g'.
Our aim is to compute
In weak coupling the higher
order terms in Eq. (8.32) are not significant.
the Action becomes
S=QI-(2(Z')Qs(s) s(s+ s)I,
(8.29)
as claimed. The one-dimensional spin system is disordered at all temperatures.
Its exponentially decaying
correlation function implies the area law and confinement for the two-dimensional gauge system. The details of this construction are essentially the same as
for the Ising gauge theory, Eq. (5.40).
It is more interesting to consider the similarities between SU(n) gauge theories in four dimensions and
Striking
SU(n) xSU(n) spin systems in two dimensions.
similarities between these two systems have been emphasized by A. M. Polyakov (1975) and A. B. Migdal
(1975). Recall some of the characteristics of the twodimensional spin systems. First, they are asymptotically free. This will be shown in the next section,
where the weak coupling renormalization group will be
obtained. Second, they cannot have a local order parameter because a continuous global symmetry cannot
break down spontaneously in two dimensions. Third,
they develop a mass gap dynamically, and their scattering amplitudes contain only massive excitations
(Zamolodchikov and Zamolodchikov, 1977). It is not
known if non-Abelian gauge theories in four dimensions
have this last property, although it is strongly hoped
for. Because of these similarities and interesting properties, it is worth our while to consider the spin systems in more detail.
Since we want to expose the geometry underlying the
asymptotic freedom of the model, we shall parametrize
the calculation judiciously.
In general each component
of s has high- and low-frequency components.
However,
if one considers only weak coupling, two components,
s f and s
an be cho sen to be s lowly varying and large,
O(l), tvhile the third component, s3, is rapidly varying
and small, O(Wg). In the course of the calculation we
shall see that this arrangement is possible. Since the
spin satisfies the constraint of Eq. (8.31) we can para. —
metrize it,
„c
s] —v 1 —s23 sing
(8.33)
s, =V'1 —s23 cosO.
Then the Lagrangian
Z
=
2g
(B„s)'=
becomes
(1 —&g)(s. ~)'+
2g
1
'2
(8.34)
If we anticipate that, for small g, s3-O(vg) we
proximate 2 by
&=(I/2g)[{s,s )'+(1 —s')(s, ~)'+s'(s, s
ca",ap-
)'+. ].
(8.35)
Finally, it is convenient to rescale s3 to eliminate the
overall factor of g,
h(r)
= s, (r)/Wg,
(8.36)
so
C. Renormalization
two dimensions
group analysis of O(n) spin systems in
It is interesting and not difficult to understand asymptotic freedom and dynamical mass generation in nonAbelian spin systems in two dimensions.
One aim of
this section is to understand asymptotic freedom in
terms of the geometry of the non-Abelian group. For
easy presentation consider the O(3) model, where one
can visualize the local spin and easily see that the local
curvature of the sphere implies asymptotic freedom.
To develop the theory's weak coupling renormalization
group, we shall integrate out the high-frequency components of the local spin variable s(n) and find an effective Action for its low-frequency components.
For
weak coupling, the effective Action will have the same
form as the original, but the effective temperature will
be increased. The partition function of the model is
Z=, , ds(s) ssp
(8.37)
This expression makes the virtues of this parametrization clear. As 0 fluctuates, it affects the contribution
of the 0 field to the Action. However, fluctuations in
6) do not react back onto h.
These facts justify the
earlier comments stating that for the purposes of weak
coupling renormalization s3 could be treated as small
and rapidly fluctuating, while s, and s~ are large and
slowly varying.
Now consider the theory with a momentum space cutoff ~p~ &A. Introduce a. high-momentum
slice, A'& ~p~
&A, and a, low-momentum
slice, 0 & p~ &A', and organize the computation of the partition function accordingly,
~
Z=
Op
1
exp —
2g
s„s s„sd'rI,
(8.30)
where the spin is constrained,
A '&@&A
(8.31)
We shall. be using continuum space-time notation
throughout this discussion, and will form a renormalization group using momentum space cutoffs as in Sec.
VII.D. Since the integrations of the renormalization
group preserve the O(3) symmetry of the model, the
Rev. Mod. Phys. , Vo(. 5'I, Na. 4, October 1979
xexp
dx
1
h p exp —— e„h d'x
2
—
2
8 8
2g
0&p&a
—
. ].
2 = —.
'[(B.h)'+ [(1/g) —h'J(s. e)'+gh'(s. h)'+
h (a 8)2d~x-
"
—
2
gh (8 h) d
(8.38)
x+
~
~ ~
For small g the second term in the last exponential can
be dropped. Then the integral over h(p) is a pure Gaussian and it can be computed exactly,
709
John B. Kogut: Lattice gauge theory and spin systems
bhh
(p) exp
'(@&A
N=exp
=X exp
I- —
1
(e„h)'d'x
1
—
(s"8)'d'x
2
p
1
h'(e„e) edex
(h')„(b, b)'d'xj
h
Ne=xp
j exp I
—
) (h'/h)
f
(e„b) d
s'-
1
d'f7
~. (2m}'
p'
(8.39)
xj,
where the overall constant N is not important.
Putting
this result back into Eq. (8.38) we see that the 8 term
of the effective Action is
&'= —.'C(1/g) —(1/2v)
In(A/A')j(8. 8) '+
high-frequency, sma, ll-amplitude fluctuations in the spin
variable s. This led to a description of the model in
terms of a new spin variable s'. Since the sphere has
local curvature, the averaging effectively reduces the
magnitude of the spin variable so that ~s'~ &1. To restore the condition that the effective spin has unit length
s'/~ s'~ which can be absorbed
requires a rescaling
into a change in the coupling constant g -g/~ s'~ ~. Therefore the effective coupling has ines erased as a result of
integrating out the high frequencies.
It is particularly interesting that the O(3) model has
significant coupling-constant renormalization even at
infinitesimal g. This occurs because even infinitesimal
fluctuations of the spin variables experience the local
curvature of the sphere. In other words, the compact
character of the group is apparent even in weak coupling. This should be contrasted with the planar model.
There small fluctuations are not sensitive to the fact
that 8(n) is an angular variable. Therefore the lowtemperature perturbative calculations are the same as
those of an ordinary free field and show no couplingconstant renormalization.
It is only at finite temperatures, where nonperturbative vortex variables are considered, that coupling-constant renormalization and a
phase transition are discovered. The relevance of the
vortex variables at sufficiently high T' is an indication
that spins are now winding through their entire orbits
and the periodicity of the 8(n) variables is affecting the
dynamics. And once the boundedness of the 0 variable
becomes important, it is clear from our intuitions
gained from the O(3) model that the planar model will
experience coupling-constant renormalization and that
its effective temperature will grow.
The generalization of Eq. (8.44} to the O(N) model is
~
. .
~
(8.40)
Comparing this with the 8 term in the original Action,
(I/2g)(B 8)'+ ~, we find coupling-constant renormaliz ation,
(1/g') = (I/g) —(I/27r) ln(A/A') .
(8.41)
~ ~
In making this identification we have used the fact discussed in Eq. (8.33) that the O(3) symmetry restricts
the form of the effective Act&on. So, although the parametrization used here obscures the O(3) symmetry of
the effective Action, the reciprocal of the coefficient of
the (8„8) term is necessarily the effective coupling constant of the effective Action equation (8.32).
It is best to convert Eq. (8.41) to a differential equation by letting the high-momentum
slice, A'&p &A, be
infinitesimal.
Setting A'= A+ 6A, we have
ln(A/A') = —ln(1+ 5A/A) = -6A/A
(8.42 a)
clear,
(1/g') —(I/g) = d(1/g) = -(1/g')dg.
Therefore
A
g
(8.42b)
1
(8.43)
(8.44)
2
2g
Equations (8.43) and (8.44) show that the theory is
asymptotically free. Equation (8.43) states that the
effective coupling is a decreasing function of the momentum cutoff. Alternatively, Eq. (8.44) means that a
smaller coupling must be used with a finer space-time
cutoff than with a coarse one. Therefore, assuming
that the effective coupling increases indefinitely with z,
Eq. (8.44) implies that the theory's continuum limit is
at g = 0. It would also mean that the theory's long-distance properties are described by a strongly coupled
Action. That Action would clearly have a finite correlation length, so the theory would be free of massless
spin waves. Other analyses suggest this point (Zamolodchikov and Zamolodchikov, 1977).
Before leaving this topic, let us discuss the physical
reason for the asymptotic freedom of this model. In
computing the effective Action we integrated out the
Rev. IVlod. Phys. , Vol.
(8.45)
It is N-
= —1 g
this in te rms of a re al space cutof f such as a
W riting
variable lattice spacing,
d~
1
a dg =
(K —2)g
51, No. 4, October 1979
2 because this is the number of directions in
which the spin can experience rapid, small fluctuations
which are Pe&Pendiculax to its average, slow motion
around the group space. Qnly the perpendicular fluc-
tuations contribute to coupling-constant renormalization. Fluctuations in the direction of the spin's average,
slow motion have the same character as fluctuations of
a planar spin model, which, as we have seen, experiences
renormalization at low temperature.
no coupling-constant
The standard notation for weak coupling renormalization equations such as Eq. (8.45) is
—p(g),
a dg
d
(8.46)
where P is the Callan —Symanzik function (Symanzik,
1970; Callan, 1970). In the language used in this discussion, positive P is asymptotic freedom.
As mentioned above, the asymptotic freedom of the
O(n) models suggests that they develop a mass gap
Recall that the planar model develops a
dynamically.
gap for temperatures above the Kosterlitz —Thouless
phase transition, and that for T = g, it behaves as
ex(p)
expI —b (
)
(8.47)
710
John B. Kogut: Lattice gauge theory and spin systems
It is easy to obtain the analogous formula for asymptotically free theories (Lane, 1974; Gross and Neveu,
1974). Simply observe that the mass gap is a physical
quantity which we can hold fixed as the cutoffs of the
underlying field theori'es are changed. Therefore
d
rn
da
=0.
Trivial indeed.
m
(8.48)
Qne can also write m in the form
= (I/a)E(g),
(8.49)
1,
by dimensional analysis.
By combining these last two
equations we can determine m as a function of g. Substituting Eq. (8.49) into Eq. (8.48) gives
—
dg
— p+(g) + +'(g)
'
da
a
Using
dg
(8.50)
Eq. (8.46) this becomes
~(g) = ~(g)/p(g),
(8.51)
which can be integrated
immediately,
(8.52)
If we choose g'« I, we can evaluate Eq. (8.52) using the
lowest order expression [Eq. (8.45)] for P(g),
x( g) = e xx (- ~2r
1
(8.53)
Therefore the mass gap measured in units of the lattice
spacing depends nonanalytically on the coupling constant. Such an effect could never be discovered using
ordinary perturbation theory. It implies that the critical singularities of such models are not the traditional
power laws. In this case the singular functions of the
critical region should be expressed as powers of the
correlation length itself .
It is interesting to consider an asymptotically free
theory which has a rich mass spectrum m„ i = 1, 2, . . . .
The O(n) spin systems are not in this class because.
they have repulsive forces. Anyway, if a richer theory's
critical point lies at g =0 and if it has no intrinsic mass
scales, then the ratios of its dynamically generated
masses rn,- must be pure numbers„
(8.54)
The mass ratios can only depend on the group of the
theory. The beauty of this result is that once a mass
scale is set, all the masses of the theory are determined with no free parameters.
It is believed that the
four-dimensional non-Abelian gauge theory of strong
interactions involves massless gauge fields and a doublet of (essentially) massless quarks. Then the lowenergy end of the hadronic mass spectrum should satisfy Eq. (8.54). Therefore many of those curious numbers of the Particle Data Table should be predicted
with no freedom 1
D. Results from the IVligdal recursion relation
How deep and reliable are the analogies between SU(n)
XSU(n) spin systems in two dimensions and SU(n) gauge
Rev. Mod. Phys. , Vom.
51, No. 4, October 1979
Exact mappings or even
approximate mappings with calculable corrections, have
not been derived. One of the obstacles to such a program has been the complexity of the link variables
U, (n) and the constraints among plaquette variables following from local gauge invariance.
It has already been
pointed out in the discussion of Abelian lattice gauge
theories that plaquette variables satisfy constraints such
a.s Eq. (5.58). The non-Abelian analogs of such constraints are much more difficult to deal with. In the
context of a renormalization group calculation, one
would begin with an Action written in terms of plaquette
variables, and integrate out various link variables to
generate an effective Action written in terms of new
plaquette variables, which satisfy the essential constraints. A practical scheme of this sort has not appeared. However, A. B. Migdal proposed some time
ago (Migdal, 1975) a recursive scheme which does
handle gauge degrees of freedom as well as spin degrees
of freedom. It is not exact and it is not known how to
calculate corrections to it. However, it is extremely
clever and applies at all values of the coupling. It produces results which in many cases agree rather well
with results obtained by other techniques.
For example, it predicts the g~ «1 coupling-constant renormalization equations of O(n) spin systems to within 30%. Its
estimates for some of the critical exponents of the twodimensional Ising model are within 10/o of the exact results. However, it has many limitations and pitfalls
which have been discussed by Kadanoff (Kadanoff, 1976b).
The Migdal scheme is particularly interesting here
because it relates gauge theories in four dimensions to
spin systems in two dimensions in just the fashion anticipated,
theories in four dimensions'P
4-D Gauge System
2-D Spin System
ZN
U(1)
U(1)
SU(N)
=
=
SU(N) &&SU(N)
The correspondence means, among other things, that
the coupling-constant renormalization problems in the
gauge and spin systems are essentially identical. Also,
the phase diagrams are related with the correspondence
confinement
—disorder
.
The SU(N)X SU(N) spin systems are predicted to have
only one phase, so the scheme predicts confinement for
non-Abelian gauge theories for all couplings 1 Detailed
relations among critical exponents also follow. For
example,
gauge
syin
(8.55)
for the mass gap exponent.
These are extremely encouraging results for a lattice
Perhaps more exacting analyses can
gauge enthusiast.
be made and will sharpen these correspondences.
IX. PARTING COMMENTS
The topics discussed in these lectures are for the
most part "old hat. " We shall close this review with a
few remarks concerning current research developments.
711
John B. Kogut: Lattice gauge theory and spin systems
A. M. Polyakov is initiating an ambitious program to
formulate and solve lattice gauge theories in three and
four dimensions by developing deep correspondences
to two-dimensional spin systems (A. M. Polyakov, 1979).
As mentioned earlier in the text, two-dimensional
O(n)
Heisenberg spin systems are soluble in the continuum
limit for n ~ 3. The reason for this lies in the fact that
which Leads to
these theories have "hidden symmetry,
an infinite number of conservation laws. In two dimensions these conservation laws prohibit particle production in scattering processes, and this simplicity leads
to solubility (Zamolodchikov and Zamolodchikov, 1977).
Such conservation Laws cannot exist in nontrivial fourdimensional theories (Coleman and Mandula, 1967).
However, Polyakov has suggested that a nontrivial extension of the conservation laws to non-Abelian pure
gauge fields in four dimensions exists and may lead to
an elegant, closed solution.
Polyakov views non-Abelian gauge theories as chiral fields defined on closed
loops in real space-time. We saw in the text that Abelian lattice gauge theory could be viewed in an analogous
fashion
the physical space of states of the theory consists of closed loops of electric flux. Recall how this
came about. The v-continuum Hamiltonian of the theory
was
"
—
0= 2a'
E~ & — &
ga
..
cos~,
„n,
(9.1a)
where 8, & is related to the magnetic field,
6» —a'gB;
(ij@ cyclic)
.
(9.lb)
The physical space of states is locally gauge invariant
or, as discussed in Sec. VI, satisfies Gauss's law.
Therefore only closed loops of electric flux are permissible. The loops must be closed because the theory
has no sources or sinks of flux. In strong coupling only
the electric term in Eq. (9.1a) is significant, and the
energy of a closed loop of flux is proportional to its
length. The magnetic term in Eq. (Q. la) allows the
loops to fluctuate but always leaves them closed. Strong
coupling expansions for the energies of such states,
"glueballs" or "boxitons, " then resemble mass gap calculations discussed earlier in the text (Kogut et af. ,
1976) .
The idea which Polyakov abstracts from this formulation of the theory is that phase factors defined on closed
paths are the basic quantities of the theory,
exp zg
Ax
~
dx
.
(9.2)
The Hamiltonian form of the non-Abelian theory leads
one to a similar perspective.
In this case we have
chiral fields defined on loops,
,', ],exp
((i/2) ag r A,.),
~
(9.3}
as the operator which generates the physical space of
locally gauge-invariant states. Next Polyakov explores
the properties of chiral fields defined on the space of
closed contours and finds intriguing similarities to
chiral fields defined on points in two-dimensional
space-time. He has proposed an infinite set of conservation laws for the gauge theories and connections
Rev. Mod. Phys. , Vol.
51, No. 4, October 1979
to fermionic string models. Some of this work is incomplete, so it is not appropriate to discuss it further
here. It is an extremely imaginative approach with
some very high goals.
In the text we limited our discussion to gauge theories
without matter fields. We discussed methods of labeling the theories' possible phases. Qf course, such
theories are unrealistic. It is believed, however, that
if the pure non-Abelian gauge theory confines quarks
then the theory with dynamical quark fields added will
have its physical spectrum exhausted by color singlets.
It is important to study this problem directly and map
out the phase diagrams of models with matter fields.
It is also most interesting to place the matter fields
in the fundamental representation of the gauge group as
in quantum chromodynamics
(Fritzsch et a/. , 1973). If
this is done, the traditional method of labeling the phase
diagram fails. In fact, the interquark potential can
never rise linearly in such theories. This is simply a
consequence of screening. Consider the theory at
strong coupling, place a static quark —antiquark pair
into the system, and separate them a distance R. Then
in the lowest-energy state dynamical quarks will materialize near the static quarks and screen their color
locally, as depicted in Fig. 40. The interquark potential then has a short range.
To study the phase diagrams of gauge theories with
matter fields, it is instructive to return to Ising Lattice
gauge theory and incorporate into it Ising matter fields
(Wegner, 1971). The degrees of freedom of this model
have the group structure of interest, and the discreteness of the variables permits easy, reliable analysis.
The theory is defined as follows: on sites there are
Ising matter fields
cr(r)
=+1,
and on links there are Ising gauge fields
V„(r) =+1.
The gauge-invariant
(9.5)
Action is
S=PQ a(r}V„(r)a(r+ p, )
U„~U„~+p,
+K
U
x+v U„r .
(9.6)
ry gkv
The local gauge symmetry sixnultaneously flips the matter field at r and all the link gauge fields emanating
from
The reader familiar with lattice versions of
will recognize the first term
quantum chromodynamics
in Eq. (9.6) as the Ising form of a gauge-invariant quark
kinetic energy.
Consider this theory's phase diagram in the (P,K)
plane (Fradkin and Shenker, 1979). Two limiting cases
r.
are easy:
(1) K=~. The gauge field is frozen, so the Action
reduces to the Ising model. For d ~ 2 this is a twophase system having (o(r)) wO for P )P, and (0(r)) =0
otherwise. The P, critical point is shown on the right-
FIG. 40. Screening the Iong-range confining potential.
John B. Kogut: Lattice gauge theory and spin systems
hand
vertical axis of Fig. 41. In field theoretic jargon,
phase of the Ising model displays the
"Higgs mechanism" because the vacuum expectation
value of the charge-bearing field is nonzero.
the magnetized
%=K, —tanh4P .
(2) P =0. Only the Ising gauge theory remains in this
case. It confines for K&K, and does not for K&K, . It
is plotted on the lower horizontal axis of Fig. 41.
The other boundaries of the phase diagram are trivial
and are free of critical points.
It is more difficult to describe the interior of the
phase diagram. First we should ask whether the critical point:s at (P, K) = (O, K,) and (P, K) = (P„~) are the endpoints of critical lines which extend into the figure.
Consider the critical point K, of the pure gauge system.
It is not hard to see that it extends into the phase diagram, because if P is small the matter fields' primary
effect is just to renormalize the temperature K, slightly
(Wegner, 1971). This is a. lattice version of vacuum polarization familiar from ordinary quantum electrodynamics. To see this, organize the calculation of the
partition function by first doing the sum over matter
fields. For small (8 this is done using high-temperature expansion methods. Define an effective Action for
the gauge fields (Fradkin and Shenker, 1979)
hhrt[s.
„[v„(r)j[=
g
v(r)=+i
ettrtIsgtrvtr+rrgvvvvI
(9 7)
Use the usual identity,
exp(Po Ua)
= coshP(1+
O'Uo'
(9.8)
tanhP)
and note that the sum over o(x) picks out only closed
loops of link variables U. The smallest closed loop is
just a plaquette, so
exrt[s„, [v (t))) e
h
e&E v
It+(tah
)h4.
S,
[vv v+v
«v e(.t ~n8) 4E v v v v
-exp SC+tanh'
UUUU
.
(9.9)
So, the effective Action is another pure gauge model
with an effective coupling,
K„,=K+tanh'P, (P«1).
(9.10)
Higgs
This theory has a critical point at K,ff
0
Free
Charge
Conf inement
K
FIG. 41. Phase diagram of Ising gauge theory
f ieIds.
Rev. Mod. Phys. , Vol.
53, No. 4, October 1979
with matter
or
(9.11)
Therefore coupling to matter fields has shifted the
transition point to a larger value of the electric charge
(K = 1/2e ) as one's physical intuition of vacuum polarization would suggest (Stack, 1978).
Similar arguments show that the spin system's transition at (P,IC) = (P„~) extends into the phase diagram.
In three-dimensional
systems this follows from the fact
that the theory is self-dual (Wegner, 1971), and in four
dimensions it follows from the fact that the theory is
dual to a higher gauge theory (Wegner, 1971) which can
be analyzed along the lines discussed above.
Although these arguments are reliable only near the
edges of the phase diagram, it is reasonable to guess
that the two critical lines meet inside the diagram as
shown in Fig. 41. Then the lower right portion of the
phase diagram would be isolated. Presumably free
charges could exist in this phase.
The other portion of the diagram is very interesting.
It is bounded on the right vertical side by the Higgs
mechanism and on the lower left horizontal side by
confinement.
Although these regions of the phase diagram differ in many quantitative aspects, one can show
that they are not separated by a critical surface. In
other words, they are continuously connected, and no
symmetry criterion or phase transition separates them.
The proof of this fact begins by observing that the theory
is trivial on the two boundaries K = 0 and P = ~. Then
convergent expansion methods imply that no critical
point can exist within a strip of finite width bordering
this wedge-shaped boundary (Fradkin and Shenker, 1970).
The free energy is analytic iri this strip, so the Higgs
mechanism and confinement phases are connected.
This result depends crucially on the fact that the matter field belongs to the fundamental representation of
the group, so that no impurities of smaller charge can
be placed into the system to measure the force law. If
the matter fields were not in the fundamental representation, then the P-~ limit of the theory would not be
trivial, and the Higgs and confinement boundaries could
be separated by a critical line (Einhorn and Savit, 1978).
Fig. 41 is the simplest phase diagram which can be
drawn that is consistent with the analysis done to date.
The results discussed here are not particular to Ising
systems. They apply equally well to lattice theories
with non-Abelian gauge groups and scalar matter fields.
Unfortunately, the extension to theories with real fermions remains unclear.
This article has devoted almost no space to one of the
major goals of lattice gauge theory the calculation of
the mass spectrum of strongly interacting particles.
The reason is that, although considerable efforts have
gone into this direction, only rather crude results have
been achieved. At present several research groups
are attempting to develop reliable calculational tools
for such projects. Strong coupling expansions, Monte
Carlo integration methods, and renormalization groups
are all being studied. Considerable dirty numerical
work will probably be necessary before a good quantitative understanding of these theories is obtained.
—
r
K
John B. Kogut: Lattice gauge theory and spin systems
Hopefully there will be considerable progress in lattice gauge theory in the next few years.
ACKNOWLEDGMENT
The author thanks the Physics Department of the University of Illinois for providing the opportunity to present a preliminary version of this article in the form of
a special topic graduate course.
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